Constraining the Type of Central Engine of GRBs with Swift Data

Two possible forms of light curves might confuse our sample selections performed through visual inspection. First, we might take an intrinsic single power-law (SPL) decay as a broken power-law decay. Second, we might also take a normal decay followed by a postjet break phase as a plateau followed by a normal decay phase.

Therefore, to characterize a plateau, we perform a temporal fit to the X-ray light curves within a time interval (t_s, t_e) for each burst. Here, t_s is the time when the plateau begins and t_e when the segment after the plateau break ends. We employ either a smoothly broken power law (BKPL),

$$\begin{eqnarray}&&F={F}_{0}\,{[{(t/{t}_{{\rm{b}}})}^{{\alpha }_{1}\omega }+{(t/{t}_{{\rm{b}}})}^{{\alpha }_{2}\omega }]}^{-1/\omega },\end{eqnarray} \tag{ 2 }$$

or an SPL⁷ to fit the light curves,

$$\begin{eqnarray}&&F={F}_{0}\,{t}^{-\alpha }.\end{eqnarray} \tag{ 3 }$$

In the above equations, α, α₁, and α₂ are the temporal slopes; t_b is the break time; F_b = F₀ 2^−1/ω is the flux of the break time; and ω describes the sharpness of the break.⁸ We perform the fits to the data with a nonlinear, least-squares fitting using the Levenberg–Marquardt algorithm and an IDL routine called mpfitfun.pro⁹ (Markwardt 2009).

To exclude intrinsic SPL decay, we first apply the above two models. We then choose the best model according to the following two principles. (i) We compare the ${\chi }_{{\rm{r}}}^{2}$ value for both models and choose the model having ${\chi }_{{\rm{r}}}^{2}$ closest to 1.¹⁰ For example, the value of the ${\chi }_{{\rm{r}}}^{2}$ of GRB 070529 for the BKPL model fitting is 29/30 (close to 1), and for the SPL model fitting it is 68/33 (much larger than 1). We then adopt the BKPL as the best model for GRB 070529. (ii) In case both models have similar ${\chi }_{{\rm{r}}}^{2}$ values (close to 1), we always chose the simplest model (SPL). For instance, GRB 101219B has ${\chi }_{{\rm{r}}}^{2}$ 16/18 (close to 1) for the BKPL model and 21/21 (close to 1) for the SPL model. In this case, the SPL model is the best model for our choice. We then remove those cases that have the SPL model as the best fit.¹¹

To properly select only plateau phases and remove the normal decay followed by a jet-break phase, we only keep the temporal decays that are too fast to be explained by an external shock. For this, we use the closure relations to obtain an estimate of the spectral index α(β) in the case of different spectral regimes (ISM/wind),

$$\begin{eqnarray}{\alpha }_{1}^{s}\lt \alpha ({\beta }_{1})\equiv \left\{\begin{array}{ll}\displaystyle \frac{3{\beta }_{1}-1}{2}, & \nu \gt \max ({\nu }_{m},{\nu }_{c}){\rm{(ISM}}\,{\rm{and}}\,{\rm{wind)}}\\ \displaystyle \frac{3{\beta }_{1}}{2}, & {\nu }_{m}\lt \nu \lt {\nu }_{c}{\rm{(ISM)}}\\ \displaystyle \frac{3{\beta }_{1}+1}{2}, & {\nu }_{m}\lt \nu \lt {\nu }_{c}{\rm{(wind)}}\end{array}\right..\end{eqnarray} \tag{ 4 }$$

Here, α₁ is the temporal decay index before the break time, Γ₁ is the photon spectral index, β₁ is the spectral index (β = Γ − 1), ν_m is the minimum frequency, and ν_c is the cooling frequency. Comparing the theoretical spectral indexes to the measured ones, we only keep the bursts that satisfy¹² ${\alpha }_{1}^{s}\lt \alpha (\beta )$.

A total of 101 GRBs are selected. These are shown in Figures 1–3.

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View extended figure (6 panels)

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In order to identify the type of plateau (internal or external), we compare the temporal (α₁ and α₂) and spectral (Γ₁ and Γ₂) indices¹³ before and after the plateau break, and combine them through closure relations. Since the external plateau is purely dynamical, no change in the spectral index is expected. As such, we require Γ₁ = Γ₂ (within errors), which is associated with the equality between the measured decay index and that obtained by closure equations from Γ₁ and Γ₂. The prebreak slope α₁ (corresponding to q = 0) is given by

$$\begin{eqnarray}{\alpha }_{1}=\left\{\begin{array}{ll}\displaystyle \frac{2{\alpha }_{2}-3}{3}, & {\nu }_{m}\lt {\nu }_{}\lt {\nu }_{c}\qquad (\mathrm{ISM})\\ \displaystyle \frac{2{\alpha }_{2}-1}{3}, & {\nu }_{m}\lt {\nu }_{}\lt {\nu }_{c}\qquad (\mathrm{Wind})\\ \displaystyle \frac{2{\alpha }_{2}-2}{3}, & {\nu }_{}\gt {\nu }_{c}\qquad (\mathrm{ISM}/\mathrm{Wind})\end{array}\right.,\end{eqnarray} \tag{ 5 }$$

while the postbreak slope α₂ (corresponding to q = 1; Zhang & Mészáros 2001), according to Zhang et al. (2006), is

$$\begin{eqnarray}{\alpha }_{2}=\left\{\begin{array}{ll}\tfrac{3{\beta }_{2}}{2}=\tfrac{3(p-1)}{4}, & {\nu }_{m}\lt {\nu }_{}\lt {\nu }_{c}\quad (\mathrm{ISM})\\ \tfrac{3{\beta }_{2}+1}{2}=\tfrac{3p-1}{4}, & {\nu }_{m}\lt {\nu }_{}\lt {\nu }_{c}\quad (\mathrm{Wind})\\ \tfrac{3{\beta }_{2}-1}{2}=\tfrac{3p-2}{4}, & {\nu }_{}\gt {\nu }_{c}\qquad (\mathrm{ISM}/\mathrm{Wind})\end{array}\right.,\end{eqnarray} \tag{ 6 }$$

where β₂ is the spectral index during the normal decay phase, and p is the electron's spectral distribution index. These criteria gave 89 bursts in this sample. Figure 4 shows all GRBs' external plateaus in the α₁−α₂ plane, with three theoretically favored lines (Equation (5)) plotted. Figure 5 shows the β−α plane based on either the ISM or the wind model. The GRBs satisfying the "closure relations" are identified as our sample GRBs with colored data points.

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The internal plateau (Troja et al. 2007; Lyons et al. 2010; Rowlinson et al. 2010, 2013) cases are selected based on Γ₁, different Γ₂, and α₂ > 2 + β. This sample is made up of only 12 bursts. The bursts that cannot be placed in either of these two categories are discarded from further analysis.¹⁴

We derive the X-ray isotropic energy released (${E}_{{\rm{X}},\mathrm{iso}}$) during the plateau phase by integrating the fluence from t_s to t_b,

$$\begin{eqnarray}&&{E}_{{\rm{X}},\mathrm{iso}}=\displaystyle \frac{4\pi {{kD}}_{{\rm{L}}}^{2}{S}_{{\rm{X}}}}{1+z},\end{eqnarray} \tag{ 7 }$$

where k = (1 + z)^β − 1, β is the spectral index, S_X is the X-ray fluence¹⁵ integrated over the plateau phase, D_L is the luminosity distance, and z is the redshift.

Another important parameter is the isotropic kinetic energy (E_K,iso), which is measured from the afterglow flux during normal decay. Here, ${E}_{{\rm{K}},\mathrm{iso}}$ is calculated following the method discussed in Zhang et al. (2007a). Since E_K,iso depends on the afterglow models (we note it here again briefly for completeness), it is important to identify the afterglow spectral regimes. We first use the observed quantity (α, β) in the normal decay phase to constrain/determine the afterglow spectral regimes of each individual burst (Li et al. 2015). We find that 18 of the 101 GRBs belong to spectral regime I (ν_x > max(ν_m, ν_c)), and 83 out of the 101 GRBs belong to spectral regime II (ν_m < ν_x < ν_c). Here, ν_x is the frequency of the X-ray band. In addition, 42 out of the 83 bursts satisfy the ISM model, and 31 out of 83 satisfy the wind model.

We consider the relations obtained by Zhang et al. (2007a) to compute the kinetic energy of the external plateau sample using the normal decay segment. Since the normal decay phase is always observed at late times (t ~ 10⁴ s) in our samples, we consider the slow cooling scenario (ν_m < ν_x < ν_c; see also Beniamini et al. 2015). In the case ν_x > max(ν_m, ν_c), the same expression can be applied to both the wind-like and the constant interstellar medium since the kinetic energy does not depend on the medium profile in this spectral regime.

$$\begin{eqnarray}{E}_{{\rm{K}},\mathrm{iso},52} & = & {\left[\displaystyle \frac{\nu {F}_{\nu }(\nu ={10}^{18}\mathrm{Hz})}{5.2\times {10}^{-14}\mathrm{erg}{{\rm{s}}}^{-1}{\mathrm{cm}}^{-2}}\right]}^{4/(p+2)}\\ & & \times {D}_{28}^{8/(p+2)}{(1+z)}^{-1}{t}_{d}^{(3p-2)/(p+2)}\\ & & \times {(1+Y)}^{4/(p+2)}{f}_{p}^{-4/(p+2)}{\varepsilon }_{B,-2}^{(2-p)/(p+2)}\\ & & \times {\varepsilon }_{e,-1}^{4(1-p)/(p+2)}{\nu }_{18}^{2(p-3)/(p+1)},\end{eqnarray} \tag{ 8 }$$

where $\nu {F}_{\nu }$ (ν = 10¹⁸ Hz) is the energy flux¹⁶ at frequency 10¹⁸ Hz in units of $\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$, t_d is the time in the observer frame in units of days, D₂₈ = D/10²⁸ is the luminosity distance in units¹⁷ of 10²⁸ cm, n is the number density in the constant ambient medium, A_* is the parameter of the stellar wind, Y is the Compton parameter, and _e and _B are the energy fraction assigned to the electron and magnetic field, respectively. f_p is a function of the power-law electron distribution p index¹⁸ (Zhang et al. 2007a),

$$\begin{eqnarray}&&{f}_{p}=6.73{\left(\displaystyle \frac{p-2}{p-1}\right)}^{p-1}{(3.3\times {10}^{-6})}^{(p-2.3)/2}.\end{eqnarray} \tag{ 9 }$$

We adopted standard values of the microphysical parameters of the shock derived from observations (e.g., Panaitescu & Kumar 2002; Yost et al. 2003), namely n = 1, Y = 1, and (_e, _B) = (0.1, 0.01).

When ${\nu }_{m}\lt \nu \lt {\nu }_{c}$, the kinetic energy depends on the type of external medium. For a wind medium, one has

$$\begin{eqnarray}{E}_{{\rm{K}},\mathrm{iso},52} & = & {\left[\displaystyle \frac{\nu {F}_{\nu }(\nu ={10}^{18}\mathrm{Hz})}{7.4\times {10}^{-14}\mathrm{erg}{{\rm{s}}}^{-1}{\mathrm{cm}}^{-2}}\right]}^{4/(p+1)}\\ & & \times {D}_{28}^{8/(p+1)}{(1+z)}^{-(p+5)/(p+1)}{t}_{d}^{(3p+1)/(p+1)}\\ & & \times {f}_{p}^{-4/(p+1)}{\varepsilon }_{B,-2}^{-1}{\varepsilon }_{e,-1}^{\tfrac{4(1+p)}{(p+1)}}\\ & & \times {A}_{* ,-1}^{-4/(p+1)}{\nu }_{18}^{2(p-3)/(p+1)},\end{eqnarray} \tag{ 10 }$$

while for a constant ISM, one has

$$\begin{eqnarray}{E}_{{\rm{K}},\mathrm{iso},52} & = & {\left[\displaystyle \frac{\nu {F}_{\nu }(\nu ={10}^{18}\mathrm{Hz})}{6.5\times {10}^{-14}\mathrm{erg}{{\rm{s}}}^{-1}{\mathrm{cm}}^{-2}}\right]}^{4/(p+3)}\\ & & \times {D}_{28}^{8/(p+3)}{(1+z)}^{-1}{t}_{d}^{3(p-1)/(p+3)}\\ & & \times {f}_{p}^{-4/(p+3)}{\varepsilon }_{B,-2}^{-\tfrac{(p+1)}{(p+3)}}{\varepsilon }_{e,-1}^{\tfrac{4(1-p)}{(p+3)}}\\ & & \times {n}^{-2/(p+3)}{\nu }_{18}^{2(p-3)/(p+3)}.\end{eqnarray} \tag{ 11 }$$

After deriving E_K,iso at the break time t_b, we next estimate the upper limit of E_K,iso(t₀) by E_K,iso(t_b). If t_s > t₀, the upper limit of E_K,iso(t₀) can be estimated by

$$\begin{eqnarray}&&{E}_{{\rm{K}},\mathrm{iso}}({t}_{0})\lt {E}_{{\rm{K}},\mathrm{iso}}({t}_{{\rm{b}}})/{\left(\displaystyle \frac{{t}_{{\rm{b}}}}{{t}_{s}}\right)}^{(1-q)}.\end{eqnarray} \tag{ 12 }$$

The error on E_K,iso(t_b) is derived by bootstrapping. For other parameters with simple mathematical expressions, the errors are calculated by error propagation.

After determining the bursts with an external plateau, we further split them into three groups. The classification is made depending on whether the X-ray isotropic energy E_X,iso and or the kinetic isotropic energy E_K,iso is larger than 2 × 10⁵² erg, corresponding to the maximum energy released by a magnetar. This is illustrated in Figure 6.

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• 1.  
  Gold Sample: bursts for which the E_X,iso of the plateau phase is greater than 2 × 10⁵² erg. The sample is made up of nine GRBs, and the power-law fits to the light curves are shown in Figure 1. This corresponds to 9% of all the bursts.
• 2.  
  Silver Sample: bursts for which the E_X,iso of the plateau phase is less than 2 × 10⁵² erg, but E_K,iso is greater than 2 × 10⁵² erg. It is also consistent with the theoretical prediction of the BH central engine model. We note that because the computation of E_K,iso relies on assumed values for the microphysical parameters, it is not possible to completely rule out the magnetar progenitor for the bursts. This sample is made up of 69 bursts, and their light curves are presented in Figure 2. This corresponds to 68% of all the bursts.
• 3.  
  Bronze Sample: bursts for which neither E_X,iso nor E_K,iso are above 2 × 10⁵² erg. For the 23 bursts in this sample, it is not possible to completely rule out the BH progenitor. Figure 3 shows the fitted light curves. This corresponds to 23% of all the bursts.

Under the assumption that a magnetar emits its energy isotropically, this classification indicates that 77% of the bursts have a BH central engine; see further discussion in Section 5.1.

The k-corrected isotropic energy released in the gamma-ray band (E_γ,iso) is retrieved from the published paper. For bursts that did not have published values, we derived them from the observed fluence (S_γ) in the detector band and k-correct to the rest-frame (1−10⁴) keV using the spectral parameters.

We obtained the fluence and spectral parameters (including the α and β spectral indexes and the peak energy E_p) of the Band function from published papers (e.g., de Pasquale et al. 2016) or the GCN Circulars Archive.¹⁹ First, if a burst was detected by Swift/BAT and also by Fermi/GBM and/or Konus/Wind, we considered the spectral parameters from either the Fermi/GBM or Konus/Wind instrument. Second, if a burst was only detected by the Swift/BAT instrument and fit with a power-law or a cutoff power-law model, we proceeded with the following steps: (1) for the cutoff power-law model, the fit parameters include α and E_p (Sakamoto et al. 2011), so we adopt the typical value of β = −2.3. (2) For the SPL model, we derive E_p from the empirical correlation log E_p = (2.76 ± 0.07) − (3.61 ± 0.26) log Γ^BAT (Zhang et al. 2007b), where Γ is the photon index of the Swift/BAT band, which was found from the Swift data archive, and adopted the typical β value of −2.3. (3). We finally test the results against the Amati relation (Amati et al. 2002), as shown in Figure 7. Our results are consistent with the Amati relation, which indicates that our method is reasonable.

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We derive the k-correction (Bloom et al. 2001) factor in the rest-frame 1−10⁴ keV band, and derive the isotropic energy E_γ,iso as

$$\begin{eqnarray}&&{E}_{\gamma ,\mathrm{iso}}=\displaystyle \frac{4\pi {{kD}}_{L}^{2}{S}_{(\gamma )}}{1+z}.\end{eqnarray} \tag{ 13 }$$

The isotropic luminosity at the break (L_X) can be derived as

$$\begin{eqnarray}&&{L}_{{\rm{X}},\mathrm{iso}}=4\pi {{kD}}_{L}^{2}{F}_{{\rm{X}},{\rm{b}}},\end{eqnarray} \tag{ 14 }$$

where F_X,b is the flux at the break time (t_b), which was obtained from the light curve fit.

The luminosity at the break time (L_X,iso) and the isotropic release energy of the plateau phase (E_X,iso), together with the time intervals (t_s, t_e) for the light curve fitting, the temporal indices of the plateau phase (α₁) and the normal decay or steeper decay phase (α₂), the break time (t_b), the X-ray fluence for the plateau phase (S_X), which is the integrated observed flux during the time interval (t_s, t_b), and the photon indices²⁰ for the shallow decay/plateau phase (Γ₁) and a normal or steeper decay phase (Γ₂), are obtained from the Swift archive,²¹ which are reported in Table 1. In addition, the table also gives the spectral indices of the plateau phase and of the following segment, both obtained from either the Swift archive or from the closure relation.

Table 1.  The X-Ray Observation Properties and Fitting Results of Our Samples

GRB	Ts−Tea	${t}_{{\rm{b}}}$ a	α1	α2	Γx,1b	Γx,2b	SXc	LX,isod	EX,isod
	(ks)	(ks)					(10−7 erg cm−2)	(1047 erg s−1)	(1050 erg)
	Gold
050401	0.13−548	5.1 ± 0.6	0.55 ± 0.01	1.47 ± 0.06	1.78 ± 0.21	1.79 ± 0.15	10.88 ± 0.57	55.73 ± 5.10	208.70 ± 10.91
050730	4.01−408	9.7 ± 0.5	0.72 ± 0.12	2.69 ± 0.04	1.43 ± 0.06	1.63 ± 0.05	11.57 ± 0.64	111.72 ± 12.58	368.39 ± 20.39
060607A	0.46−171	12.6 ± 0.3	0.44 ± 0.02	3.60 ± 0.09	1.51 ± 0.10	1.56 ± 0.08	12.91 ± 0.17	25.64 ± 1.27	273.82 ± 3.52
061222A	0.21−1443	47.6 ± 3.1	0.70 ± 0.01	1.74 ± 0.03	1.83 ± 0.06	2.07 ± 0.09	20.18 ± 0.39	4.70 ± 0.34	220.53 ± 4.25
080721	0.11−1396	3.1 ± 0.2	0.80 ± 0.01	1.65 ± 0.01	1.81 ± 0.02	1.96 ± 0.06	40.29 ± 0.68	239.60 ± 12.48	644.10 ± 10.93
100902A	1.19−1489	839.9 ± 48.9	0.73 ± 0.02	4.53 ± 0.00	2.28 ± 0.16	2.38 ± 0.54	6.93 ± 0.10	0.88 ± 0.11	268.32 ± 3.79
111209A	0.43−49	18.4 ± 0.1	0.81 ± 0.00	5.64 ± 0.05	1.24 ± 0.01	1.54 ± 0.05	288.62 ± 0.33	7.79 ± 0.05	364.77 ± 0.41
140206A	0.42−1316	13.6 ± 1.9	0.73 ± 0.02	1.41 ± 0.02	1.68 ± 0.09	1.81 ± 0.06	15.34 ± 0.72	21.01 ± 3.23	265.97 ± 12.53
150403A	0.08−2834	2.5 ± 0.1	0.50 ± 0.01	1.45 ± 0.01	1.67 ± 0.01	1.75 ± 0.04	55.11 ± 0.58	279.55 ± 5.79	588.12 ± 6.20
	Silver
050315	5.37−888	231.4 ± 29.4	0.68 ± 0.03	1.94 ± 0.16	1.89 ± 0.07	2.16 ± 0.21	6.64 ± 0.29	0.37 ± 0.05	64.21 ± 2.83
050319	0.44−1389	36.8 ± 10.8	0.49 ± 0.04	1.56 ± 0.20	1.92 ± 0.08	2.17 ± 0.23	3.40 ± 0.41	5.43 ± 0.88	78.17 ± 9.38
050505	2.89−1127	26.1 ± 3.2	0.77 ± 0.05	1.80 ± 0.06	1.98 ± 0.08	2.00 ± 0.10	3.09 ± 0.17	10.91 ± 1.72	111.58 ± 6.13
050802	0.32−830	6.6 ± 0.7	0.66 ± 0.03	1.66 ± 0.04	1.76 ± 0.09	1.90 ± 0.09	3.46 ± 0.16	4.32 ± 0.60	26.44 ± 1.21
050803	0.42−1329	16.3 ± 1.3	0.39 ± 0.03	1.82 ± 0.06	1.81 ± 0.10	2.56 ± 0.24	3.58 ± 0.16	0.10 ± 0.01	1.71 ± 0.08
060115	5.37−418	39.9 ± 27.0	0.60 ± 0.18	1.46 ± 0.22	2.16 ± 0.19	2.15 ± 0.29	0.42 ± 0.13	0.97 ± 0.68	11.19 ± 3.42
060210	3.83−1581	28.4 ± 10.4	0.83 ± 0.07	1.41 ± 0.05	2.02 ± 0.06	1.97 ± 0.09	5.31 ± 0.82	13.82 ± 5.64	165.07 ± 25.61
060418	0.36−537	1.3 ± 0.4	0.87 ± 0.13	1.57 ± 0.05	1.94 ± 0.21	1.91 ± 0.23	1.85 ± 0.36	13.46 ± 5.73	10.94 ± 2.15
060502A	0.33−1581	30.0 ± 9.2	0.55 ± 0.04	1.19 ± 0.06	1.91 ± 0.14	1.84 ± 0.17	2.29 ± 0.27	0.44 ± 0.11	13.88 ± 1.65
060604	1.24−767	22.3 ± 5.5	0.40 ± 0.09	1.27 ± 0.06	2.07 ± 0.16	2.07 ± 0.17	0.73 ± 0.10	0.77 ± 0.17	8.28 ± 1.13
060605	0.20−127	8.7 ± 0.6	0.46 ± 0.03	2.19 ± 0.08	1.97 ± 0.12	2.08 ± 0.15	1.45 ± 0.05	13.47 ± 1.18	42.73 ± 1.43
060614	1.00−2770	50.2 ± 2.6	0.12 ± 0.04	1.97 ± 0.04	1.73 ± 0.10	1.83 ± 0.12	3.02 ± 0.12	(2.12 ± 0.14)e–3	(11.71 ± 0.48)e−2
060714	0.30−1245	4.6 ± 1.6	0.43 ± 0.11	1.31 ± 0.05	1.81 ± 0.15	2.00 ± 0.18	1.03 ± 0.20	9.04 ± 3.17	17.58 ± 3.36
060729	0.40−11845	65.8 ± 2.2	0.15 ± 0.02	1.42 ± 0.01	2.04 ± 0.04	2.03 ± 0.05	13.08 ± 0.31	0.18 ± 0.01	10.41 ± 0.25
060814	1.00−1325	13.2 ± 1.8	0.51 ± 0.05	1.43 ± 0.03	2.00 ± 0.11	2.15 ± 0.09	3.35 ± 0.24	0.57 ± 0.08	6.55 ± 0.47
060906	0.33−258	12.6 ± 1.9	0.29 ± 0.06	1.82 ± 0.17	2.07 ± 0.21	1.77 ± 0.25	0.54 ± 0.05	2.52 ± 0.38	15.27 ± 1.33
060908	0.08−450	0.7 ± 0.2	0.44 ± 0.10	1.56 ± 0.07	2.03 ± 0.29	1.97 ± 0.22	1.41 ± 0.21	34.19 ± 9.41	12.86 ± 1.95
061121	0.30−2097	6.7 ± 0.5	0.41 ± 0.03	1.48 ± 0.02	1.95 ± 0.12	1.82 ± 0.06	9.55 ± 0.43	7.68 ± 0.72	44.70 ± 2.02
070129	1.15−1490	21.9 ± 3.6	0.24 ± 0.06	1.20 ± 0.04	2.28 ± 0.19	2.13 ± 0.18	1.06 ± 0.12	1.80 ± 0.23	14.11 ± 1.55
070306	0.50−1036	30.3 ± 1.7	0.13 ± 0.03	1.88 ± 0.05	1.79 ± 0.09	2.11 ± 0.18	6.77 ± 0.27	2.72 ± 0.14	40.47 ± 1.64
070508	0.08−646	0.9 ± 0.1	0.48 ± 0.02	1.43 ± 0.01	1.77 ± 0.04	1.74 ± 0.07	9.47 ± 0.33	19.66 ± 1.08	17.62 ± 0.62
070529	0.17−445	1.7 ± 0.8	0.64 ± 0.13	1.29 ± 0.05	1.75 ± 0.21	2.10 ± 0.30	0.70 ± 0.14	12.95 ± 6.48	10.52 ± 2.16
080310	1.35−378	10.7 ± 0.9	0.13 ± 0.09	1.63 ± 0.06	1.96 ± 0.17	1.82 ± 0.14	0.87 ± 0.06	2.77 ± 0.26	12.39 ± 0.91
080430	0.30−2879	29.4 ± 4.7	0.39 ± 0.03	1.15 ± 0.03	2.05 ± 0.11	2.06 ± 0.13	1.94 ± 0.16	0.11 ± 0.01	3.16 ± 0.26
080516	0.14−47	4.2 ± 2.1	0.32 ± 0.10	1.10 ± 0.12	2.40 ± 0.57	2.21 ± 0.40	0.82 ± 0.22	16.04 ± 6.10	18.51 ± 4.97
080605	0.10−230	0.5 ± 0.0	0.52 ± 0.04	1.37 ± 0.02	1.67 ± 0.04	1.71 ± 0.07	4.61 ± 0.37	113.13 ± 11.75	32.62 ± 2.64
080905B	0.20−895	4.2 ± 1.1	0.28 ± 0.11	1.46 ± 0.04	1.53 ± 0.16	2.11 ± 0.12	3.80 ± 0.62	32.05 ± 10.09	51.92 ± 8.47
081008	0.52−261	18.5 ± 4.7	0.86 ± 0.04	1.85 ± 0.14	2.05 ± 0.13	1.83 ± 0.32	3.88 ± 0.24	1.48 ± 0.46	38.19 ± 2.39
081221	0.26−506	0.6 ± 0.1	0.25 ± 0.12	1.32 ± 0.02	1.99 ± 0.07	2.04 ± 0.11	1.98 ± 0.35	219.92 ± 22.88	24.91 ± 4.36
090407	5.47−978	120.7 ± 17.4	0.59 ± 0.04	1.95 ± 0.14	2.24 ± 0.13	1.84 ± 0.37	3.05 ± 0.19	0.14 ± 0.02	17.12 ± 1.06
090418A	0.14−216	3.1 ± 0.5	0.49 ± 0.06	1.57 ± 0.04	2.02 ± 0.22	1.96 ± 0.10	3.92 ± 0.31	11.59 ± 2.50	26.74 ± 2.14
090510	0.10−67	1.4 ± 0.2	0.62 ± 0.04	2.15 ± 0.09	1.67 ± 0.10	2.05 ± 0.20	3.28 ± 0.19	5.01 ± 0.77	7.39 ± 0.42
090516	3.89−521	16.9 ± 2.8	0.77 ± 0.09	1.86 ± 0.08	2.05 ± 0.09	2.16 ± 0.12	1.68 ± 0.17	14.99 ± 3.36	56.37 ± 5.73
090529	6.85−774	234.1 ± 118.0	0.51 ± 0.10	1.74 ± 0.54	1.90 ± 0.44	1.66 ± 0.58	0.41 ± 0.08	0.03 ± 0.02	6.67 ± 1.27
090530	0.38−791	41.6 ± 15.6	0.54 ± 0.04	1.30 ± 0.13	2.03 ± 0.15	1.93 ± 0.32	1.15 ± 0.16	0.11 ± 0.03	5.02 ± 0.69
090618	0.40−2841	8.4 ± 0.5	0.72 ± 0.01	1.51 ± 0.01	1.93 ± 0.04	1.86 ± 0.05	24.04 ± 0.53	1.34 ± 0.08	19.13 ± 0.42
091018	0.07−520	0.5 ± 0.1	0.31 ± 0.06	1.24 ± 0.02	1.84 ± 0.15	2.04 ± 0.09	1.55 ± 0.17	11.60 ± 1.57	4.05 ± 0.45
100418A	0.40−2091	125.0 ± 17.1	−0.01 ± 0.05	1.66 ± 0.10	1.89 ± 0.29	1.84 ± 0.17	2.09 ± 0.23	(2.26 ± 0.31)e–2	2.24 ± 0.24
100615A	0.20−163	19.8 ± 4.5	0.39 ± 0.03	1.26 ± 0.10	2.24 ± 0.19	1.92 ± 0.25	12.82 ± 1.44	4.10 ± 0.60	67.44 ± 7.57
100621A	0.40−1769	99.3 ± 20.6	0.86 ± 0.03	1.66 ± 0.08	2.33 ± 0.10	3.06 ± 0.31	25.66 ± 0.99	0.11 ± 0.03	20.58 ± 0.80
100704A	0.50−1051	33.3 ± 6.4	0.68 ± 0.03	1.38 ± 0.05	2.04 ± 0.10	1.92 ± 0.17	5.10 ± 0.32	6.15 ± 1.14	139.15 ± 8.85
100814A	1.00−4848	148.9 ± 7.3	0.50 ± 0.02	2.09 ± 0.06	1.90 ± 0.05	2.08 ± 0.13	12.25 ± 0.28	0.58 ± 0.03	68.13 ± 1.58
100906A	0.30−202	13.2 ± 1.3	0.76 ± 0.03	2.12 ± 0.80	1.96 ± 0.10	2.05 ± 0.15	11.20 ± 0.35	3.73 ± 0.53	59.73 ± 1.86
111008A	0.34−1014	7.4 ± 0.9	0.29 ± 0.04	1.34 ± 0.03	1.92 ± 0.09	1.88 ± 0.13	2.57 ± 0.22	52.83 ± 6.48	116.40 ± 9.81
111123A	4.22−183	42.9 ± 12.3	0.81 ± 0.08	2.02 ± 0.29	2.23 ± 0.20	2.31 ± 0.47	0.99 ± 0.10	1.44 ± 0.52	21.80 ± 2.28
111228A	0.50−2572	6.5 ± 0.8	0.22 ± 0.05	1.23 ± 0.02	2.01 ± 0.15	2.15 ± 0.10	3.29 ± 0.30	1.00 ± 0.11	4.63 ± 0.42
120327A	0.30−146	3.0 ± 0.6	0.60 ± 0.06	1.51 ± 0.06	1.69 ± 0.14	1.82 ± 0.17	1.97 ± 0.20	20.09 ± 4.75	35.78 ± 3.65
120712A	0.10−67	4.6 ± 1.5	0.89 ± 0.04	1.64 ± 0.10	2.01 ± 0.15	2.25 ± 0.20	1.12 ± 0.08	15.03 ± 6.22	36.00 ± 2.55
120811C	0.30−81	1.7 ± 0.8	0.36 ± 0.15	1.16 ± 0.09	1.78 ± 0.15	2.24 ± 0.21	0.86 ± 0.28	34.08 ± 11.77	14.41 ± 4.75
120922A	0.74−751	3.4 ± 1.6	0.43 ± 0.19	1.09 ± 0.04	2.20 ± 0.24	1.98 ± 0.15	0.87 ± 0.31	18.93 ± 8.11	18.53 ± 6.56
121024A	3.90−236	29.7 ± 12.7	0.75 ± 0.11	1.67 ± 0.18	1.93 ± 0.14	1.58 ± 0.27	1.07 ± 0.19	0.49 ± 0.23	13.87 ± 2.48
121128A	0.20−70	1.6 ± 0.2	0.52 ± 0.07	1.68 ± 0.04	1.91 ± 0.11	1.97 ± 0.13	2.91 ± 0.26	43.37 ± 7.23	34.86 ± 3.13
121211A	3.94−362	36.1 ± 16.6	0.71 ± 0.09	1.51 ± 0.19	1.94 ± 0.12	2.27 ± 0.44	1.46 ± 0.27	0.15 ± 0.07	4.21 ± 0.79
130420A	0.76−1295	65.4 ± 29.2	0.72 ± 0.03	1.27 ± 0.09	2.13 ± 0.10	1.96 ± 0.23	1.98 ± 0.24	0.10 ± 0.04	9.03 ± 1.08
130606A	5.32−238	13.0 ± 3.6	0.44 ± 0.30	1.82 ± 0.14	1.79 ± 0.17	1.85 ± 0.21	0.57 ± 0.16	15.88 ± 5.53	32.93 ± 9.38
130612A	0.76−44	3.0 ± 2.1	0.38 ± 0.36	1.24 ± 0.16	2.19 ± 0.40	2.11 ± 0.43	0.09 ± 0.04	0.95 ± 0.60	0.91 ± 0.46
131030A	0.36−1661	3.4 ± 1.1	0.81 ± 0.05	1.28 ± 0.02	1.71 ± 0.04	1.95 ± 0.09	6.88 ± 0.91	9.39 ± 3.26	31.22 ± 4.13
131105A	0.41−734	4.8 ± 1.3	0.23 ± 0.13	1.20 ± 0.05	1.90 ± 0.17	1.99 ± 0.18	1.04 ± 0.20	3.32 ± 0.92	7.74 ± 1.50
140512A	0.23−264	17.7 ± 1.7	0.79 ± 0.01	1.68 ± 0.05	1.76 ± 0.07	1.90 ± 0.09	24.66 ± 0.64	0.96 ± 0.10	35.81 ± 0.94
140518A	0.29−19	2.6 ± 0.6	0.19 ± 0.10	1.53 ± 0.18	1.99 ± 0.15	1.94 ± 0.34	0.43 ± 0.07	28.04 ± 4.55	17.74 ± 2.88
140703A	3.81−79	14.5 ± 1.6	0.60 ± 0.10	2.28 ± 0.10	1.78 ± 0.10	1.94 ± 0.15	3.66 ± 0.30	16.93 ± 2.78	80.03 ± 6.66
141121A	56.28−1393	358.5 ± 51.9	0.44 ± 0.12	2.47 ± 0.26	1.79 ± 0.23	1.72 ± 0.21	2.58 ± 0.24	0.06 ± 0.01	14.93 ± 1.39
150910A	0.20−224	7.6 ± 0.4	0.57 ± 0.01	2.36 ± 0.06	1.83 ± 0.20	1.75 ± 0.08	16.07 ± 0.32	8.86 ± 0.57	80.16 ± 1.59
151027A	0.40−908	4.4 ± 0.2	0.22 ± 0.02	1.68 ± 0.02	2.03 ± 0.04	2.01 ± 0.06	17.07 ± 0.50	9.88 ± 0.47	30.99 ± 0.91
160121A	0.20−46	20.8 ± 4.9	0.30 ± 0.05	2.14 ± 0.77	1.97 ± 0.23	1.85 ± 0.66	1.20 ± 0.15	0.84 ± 0.14	11.76 ± 1.42
160227A	1.00−1647	21.2 ± 3.7	0.24 ± 0.09	1.22 ± 0.03	1.73 ± 0.09	1.73 ± 0.10	3.06 ± 0.36	3.43 ± 0.56	42.01 ± 4.91
160327A	0.30−42	4.8 ± 1.9	0.40 ± 0.15	1.60 ± 0.14	1.71 ± 0.22	2.03 ± 0.23	0.45 ± 0.09	16.51 ± 7.14	20.33 ± 4.04
161117A	1.05−1365	7.5 ± 1.4	0.36 ± 0.11	1.17 ± 0.03	2.11 ± 0.86	1.95 ± 0.08	2.17 ± 0.29	3.42 ± 0.57	13.82 ± 1.86
170113A	0.20−877	6.0 ± 1.2	0.57 ± 0.03	1.26 ± 0.03	1.83 ± 0.11	1.81 ± 0.09	3.77 ± 0.32	7.02 ± 1.36	37.13 ± 3.17
	Bronze
051109B	0.18−158	3.3 ± 0.8	0.26 ± 0.10	1.31 ± 0.08	2.22 ± 0.33	1.86 ± 0.34	0.23 ± 0.04	(8.38 ± 1.73)e–4	(3.63 ± 0.57)e−3
051221A	3.00−848	26.7 ± 5.2	0.18 ± 0.15	1.43 ± 0.06	1.82 ± 0.22	1.96 ± 0.15	0.39 ± 0.07	(1.77 ± 0.34)e–2	0.32 ± 0.06
060526	1.00−314	97.1 ± 20.7	0.63 ± 0.60	2.89 ± 0.48	1.91 ± 0.16	1.91 ± 0.45	1.22 ± 0.08	0.38 ± 0.14	27.66 ± 1.77
060708	0.20−1228	8.8 ± 2.2	0.59 ± 0.04	1.32 ± 0.05	2.10 ± 0.18	2.08 ± 0.16	0.88 ± 0.08	(8.73 ± 2.07)e–3	0.14 ± 0.01
061021	4.20−4236	29.7 ± 13.9	0.75 ± 0.07	1.19 ± 0.03	1.94 ± 0.08	1.87 ± 0.08	2.81 ± 0.61	0.02 ± 0.01	0.89 ± 0.20
061110A	3.00−756	73.2 ± 51.6	0.19 ± 0.27	1.16 ± 0.25	2.11 ± 0.72	1.65 ± 0.46	0.14 ± 0.06	(3.56 ± 2.06)e–3	0.23 ± 0.10
061201	0.11−120	2.6 ± 0.5	0.53 ± 0.09	2.01 ± 0.11	1.33 ± 0.18	2.01 ± 0.40	1.36 ± 0.11	(8.83 ± 2.57)e–3	(4.13 ± 0.34)e−2
070110	4.08−29	20.4 ± 0.2	0.03 ± 0.05	9.39 ± 0.69	2.07 ± 0.08	2.09 ± 0.19	3.23 ± 0.04	2.75 ± 0.11	20.55 ± 0.24
080707	0.11−279	14.4 ± 4.3	0.36 ± 0.05	1.29 ± 0.10	2.05 ± 0.30	1.83 ± 0.23	0.47 ± 0.07	0.15 ± 0.04	1.95 ± 0.29
081007	0.30−1534	68.9 ± 28.9	0.75 ± 0.03	1.35 ± 0.10	2.01 ± 0.13	2.03 ± 0.21	2.66 ± 0.26	0.01 ± 0.01	2.03 ± 0.20
081029	2.76−268	18.2 ± 1.2	0.43 ± 0.06	2.70 ± 0.15	1.92 ± 0.10	2.04 ± 0.21	1.03 ± 0.05	6.24 ± 0.61	31.11 ± 1.42
100219A	5.00−121	26.4 ± 3.6	0.24 ± 0.31	2.96 ± 0.83	1.66 ± 0.30	1.48 ± 0.39	1.70 ± 0.18	5.26 ± 1.28	69.43 ± 7.38
100302A	1.29−1062	17.0 ± 11.5	0.30 ± 0.14	0.92 ± 0.08	1.88 ± 0.35	1.95 ± 0.22	0.17 ± 0.07	1.61 ± 0.72	7.12 ± 2.92
100425A	0.34−498	25.3 ± 13.7	0.48 ± 0.06	1.16 ± 0.14	2.22 ± 0.26	1.91 ± 0.39	0.43 ± 0.09	0.17 ± 0.07	3.43 ± 0.74
110715A	0.10−847	0.2 ± 0.0	0.20 ± 0.12	1.00 ± 0.01	1.87 ± 0.10	1.83 ± 0.09	1.84 ± 0.40	35.25 ± 3.79	3.42 ± 0.74
110808A	4.00−617	40.9 ± 37.2	0.33 ± 0.20	1.07 ± 0.19	2.54 ± 0.49	1.61 ± 0.39	0.35 ± 0.18	0.05 ± 0.03	1.71 ± 0.87
120422A	0.30−1703	164.6 ± 69.5	0.27 ± 0.07	1.27 ± 0.21	2.09 ± 0.26	1.61 ± 0.29	0.22 ± 0.05	(1.98 ± 0.61)e–4	0.05 ± 0.01
120521C	0.57−42	21.2 ± 5.2	0.31 ± 0.10	2.55 ± 0.80	1.82 ± 0.32	2.19 ± 0.77	0.25 ± 0.03	4.35 ± 1.06	14.92 ± 1.86
130408A	9.55−79	27.2 ± 2.7	0.55 ± 0.21	3.87 ± 0.42	2.26 ± 0.19	2.53 ± 0.25	1.04 ± 0.09	11.83 ± 2.66	30.26 ± 2.63
130603B	0.07−118	7.6 ± 1.3	0.68 ± 0.05	1.90 ± 0.12	1.94 ± 0.16	1.97 ± 0.28	2.23 ± 0.12	0.04 ± 0.01	0.75 ± 0.04
140903A	0.21−111	8.9 ± 1.7	0.10 ± 0.07	1.28 ± 0.09	1.70 ± 0.21	1.56 ± 0.20	0.93 ± 0.13	(3.15 ± 0.46)e–2	0.30 ± 0.04
151215A	0.30−175	1.9 ± 2.0	0.54 ± 0.20	1.17 ± 0.14	2.23 ± 0.22	1.91 ± 0.55	0.22 ± 0.11	3.92 ± 3.47	3.42 ± 1.76
161108A	0.65−645	41.6 ± 23.3	0.24 ± 0.09	0.94 ± 0.13	1.78 ± 0.25	1.76 ± 0.32	0.72 ± 0.24	0.08 ± 0.02	2.65 ± 0.87

Notes.

^aThe start, end, and break times for the light curve fits. ^bThe X-ray photo indices before and after the break time. ^cThe fluence during the plateau phase, calculated by integrating the fitting light curve from start to break time. ^dThe luminosity at the break time, in units of 10⁴⁷ erg s⁻¹. The X-ray release energy during plateau phases, which is calculated by integrating the luminosity during the time interval of the plateau phase, in units of 10⁵⁰ erg.

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In this section, we study the beaming correction with three methods and compare their results. The beaming correction is defined as

$$\begin{eqnarray}&&{f}_{{\rm{b}}}=1-\cos {\theta }_{j}\simeq (1/2){\theta }_{j}^{2}.\end{eqnarray} \tag{ 15 }$$

It is used to correct the γ-ray and kinetic energies. We obtain the geometrically corrected gamma-ray energy through E_γ = f_bE_γ,iso, and E_K = f_bE_K,iso. The latter quantity is used for collimated outflows from a BH central engine (see Section 3.2).

The jet-opening angle can be derived from observational data (Rhoads 1999; Sari et al. 1999; Frail et al. 2001),

$$\begin{eqnarray}{\theta }_{j} & = & 0.124{\left(\displaystyle \frac{{t}_{j}}{1\mathrm{day}}\right)}^{3/8}\\ & & \times {\left(\displaystyle \frac{1+z}{2}\right)}^{-3/8}{\left(\displaystyle \frac{{E}_{{\rm{k}},\mathrm{iso}}}{{10}^{52}\mathrm{erg}}\right)}^{-1/8}{\left(\displaystyle \frac{n}{1{\mathrm{cm}}^{-3}}\right)}^{1/8},\end{eqnarray} \tag{ 16 }$$

where ${\theta }_{j}$ is the jet-break time, E_K,iso is the isotropic kinetic energy, and n is the medium density. If the light curve shows a jet break, we measure θ_j using Equation (16). In our sample, 18 GRBs show a jet break. For the rest of the GRBs without a jet break, we use the observed time of the last data point as the jet-break time (t_j) to estimate θ_j and get an upper limit.

Another method to estimate the jet-opening angle is by empirical relations (Nemmen 2012), θ_j ≈ 5.0/Γ₀, where Γ₀ is the initial Lorentz factor, also derived using the empirical relation as

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{0}=27{E}_{\gamma ,\mathrm{iso},52}^{0.26},\end{eqnarray} \tag{ 17 }$$

where E_γ,iso,52 is the isotropic energy, in units of 10⁵² erg.

Figure 8 shows the distributions of the beaming factors f_b. We compare the distributions of these two methods. These distributions can be fitted with Gaussian functions; we have log f_b = −1.68 ± 0.61 from using the late data point method and log f'_b = −2.14 ± 0.47 from the empirical relation method. We find that the f_b values from the first method is greater than those from the second method, and the f_b of θ_j = 5^o lies between the two distributions. In Nemmen (2012), a coefficient of 5 was adopted for the estimation. In turn, if it is initially defined as θ_j ≡ a_j/Γ₀, we then can estimate a_j. Figure 8(b) shows the distribution of a_j, with the best Gaussian fit a_j = 5.11 ± 7.21. This result is consistent with the finding in Nemmen (2012). The results, which include θ_j, f_b, a_j, ${\theta }_{j}^{{\prime} }$, and ${f}_{{\rm{b}}}^{{\prime} }$, are presented in Table 2.

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We finally adopted the following criterion to calculate the beaming factor: if a burst has an observed light curve jet break, we then use it to estimate θ_j; otherwise, we adopt θ_j = 5^o.

Another point we want to investigate is whether the γ-ray radiative efficiency is different among the samples. The radiative efficiency is defined as (Lloyd-Ronning & Zhang 2004)

$$\begin{eqnarray}&&{\eta }_{\gamma }=\displaystyle \frac{{E}_{\gamma ,\mathrm{iso}}}{{E}_{\gamma ,\mathrm{iso}}+{E}_{{\rm{K}},\mathrm{iso}}}=\displaystyle \frac{{E}_{\gamma }}{{E}_{\gamma }+{E}_{{\rm{K}}}}.\end{eqnarray} \tag{ 18 }$$

Since the kinetic energy (${E}_{{\rm{K}},\mathrm{iso}}$) increases with time during the plateau phase but remains the same during the normal decay phase, the radiation efficiency η_γ thus depends on which E_K epoch is adopted. Here we focus on the radiation efficiency at the end of the shallow decay phase, η_γ(t_b). The physical meaning for this is the efficiency of converting the spin-down energy of a BH/magnetar to γ-ray radiation.

The X-ray efficiency can also be defined as the conversion of the BH/magnetar spin-down energy to radiation, i.e.,

$$\begin{eqnarray}&&{\eta }_{{\rm{X}}}=\displaystyle \frac{{L}_{{\rm{X}},\mathrm{iso}}}{{L}_{{\rm{X}},\mathrm{iso}}+{L}_{{\rm{K}},\mathrm{iso}}}=\displaystyle \frac{{L}_{{\rm{X}}}}{{L}_{{\rm{X}}}+{L}_{{\rm{K}}}}.\end{eqnarray} \tag{ 19 }$$

In Figure 9, we compare E_γ,iso and E_K,iso (Figure 9(a)) and L_X,iso and L_K,iso (Figure 9(b)) for different samples. The same distribution is also shown in Figures 10(a) and (b). The typical value is log ${\eta }_{\gamma }({t}_{{\rm{b}}})=-0.53\,\pm \,0.56$ for ${\eta }_{\gamma }$ and is log η_x(t_b) = −0.54 ± 0.58 for η_x. The radiative efficiencies in the γ- and X-ray energy bands, in general, present the same distribution.

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One of the leading models for the GRB central engine is a stellar-mass BH surrounded by a hyper-accreting disk (e.g., Popham et al. 1999; Narayan et al. 2001; Frank et al. 2002; Kato et al. 2008; Abramowicz & Fragile 2013; Lei et al. 2013; Blaes 2014; Yuan & Narayan 2014), with a typical accretion rate of 0.01–1 M_⊙ s⁻¹. There are two main energy reservoirs that provide the jet power, namely the accretion energy in the disk that is carried by neutrinos and antineutrinos, which annihilate each other and power a bipolar outflow (e.g., di Matteo et al. 2002; Kohri & Mineshige 2002; Gu et al. 2006; Chen & Beloborodov 2007; Janiuk et al. 2007; Liu et al. 2007; Lei et al. 2008, 2009), and the spin energy of the BH, which can be tapped through magnetic fields via the BZ (Blandford & Znajek 1977) mechanism (see also Lee et al. 2000; Li 2000; Lei et al. 2013).

As suggested by Lei et al. (2013), the jet might be dominated by BZ power especially at late times (Fan & Wei 2005; Zhang & Yan 2011). The rotational energy of a BH with angular momentum ${J}_{\bullet }$ is a fraction of the BH mass M_• (or dimensionless mass m_• = M_•/M_⊙),

$$\begin{eqnarray}&&{E}_{\mathrm{rot}}={f}_{\mathrm{rot}}({a}_{\bullet })\displaystyle \frac{{M}_{\bullet }}{{M}_{\odot }}{c}^{2}\mathrm{erg}=1.8\times {10}^{54}{f}_{\mathrm{rot}}({a}_{\bullet })\displaystyle \frac{{M}_{\bullet }}{{M}_{\odot }}\mathrm{erg},\end{eqnarray} \tag{ 20 }$$

where ${a}_{\bullet }={J}_{\bullet }c/({{GM}}_{\bullet }^{2})$ is the BH spin parameter and

$$\begin{eqnarray}&&{f}_{\mathrm{rot}}({a}_{\bullet })=1-\sqrt{(1+\sqrt{1-{a}_{\bullet }^{2}})/2}.\end{eqnarray} \tag{ 21 }$$

We then connect the kinetic energy of the jet to the spin energy of the BH through the relation

$$\begin{eqnarray}&&\eta {E}_{\mathrm{rot}}={f}_{{\rm{b}}}{E}_{{\rm{K}},\mathrm{iso}},\end{eqnarray} \tag{ 22 }$$

where η is the efficiency of converting the spin energy to the kinetic energy of the jet through the BZ mechanism, and f_b is the beaming factor (see Section 2.5 for a detailed discussion). For a maximally rotating BH, about 31% of the rotational energy can be taken out to power a GRB (Lee et al. 2000). This efficiency is not sensitive to the BH spin a_•. Therefore, we take η  0.3 in the following.

From Equations (20) and (22), one can constrain the BH spin parameter once we know the BH mass M_•, the jet-opening angle θ_j, and the kinetic energy of the jet E_K,iso. Following Kochanek (2015), the typical value for the BH mass in a long GRB is 6 M_⊙. In this paper, we adopt M_• = 3 M_⊙ for short GRBs²² and 6 M_⊙ for long GRBs.

For a_• > 0.1, the bimodal distribution of the index q can be explained by the spin evolution of the BH. The BH evolves with time during a GRB, since it would be spun up by accretion and spun down by the BZ mechanism (Lei et al. 2017). Due to the different initial parameter sets for different GRBs, the evolution of the characteristics varies from burst to burst. Therefore, we have q > 0 for the spin-down evolution and q < 0 for the spin-up case.

For a_• < 0.1, the BH spin does not necessarily need to be very low. Instead, the value we obtained based on Equation (22) should be the change in the BH spin Δa_• during a GRB. The very small value of Δa_• means that the BH spin does not evolve. This is possible when the BH spin is at a critical value where either the spin-up due to accretion is comparable to the spin-down due to the BZ mechanism or the accretion rate is quite low. A clear evidence comes from the distribution of q when a_• < 0.1, for which q is close to zero.

The correlation a_•−q for our different samples is presented in Figure 11. We find that the Gold sample bursts cluster into the region where a_• > 0.1. The results are consistent with the prediction of the BH central engine model. Here we note that since the Bronze sample bursts could still have a BH central engine, we also calculate a_• for them. Interestingly, we found all Bronze sample bursts have a_• < 0.1.

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The BH parameter (a_•), the duration of the prompt emission T₉₀, the isotropic energy release in the gamma-ray band E_γ,iso, the isotropic kinetic energy E_K,iso, the electron spectral index p, the luminosity inject index q, and a redshift z (searched from published papers or the GCN Circulars Archive) are reported in Table 3 for the Gold and the Silver samples.

Table 2.  The Results of the Estimation of the Jet-opening Angle

GRB	θja	fba	ajb	${\theta }_{j}^{{\prime} }$ c	${f}_{{\rm{b}}}^{{\prime} }$ c
	(10−2)	(10−2)		(10−2)	(10−2)
	Gold
050401	15.09 ± 0.62	1.14 ± 0.09	10.79	6.99 ± 0.39	0.24 ± 0.03
050730	⋯	⋯	⋯	59.76 ± 1.62	17.36 ± 0.91
060607A	⋯	⋯	⋯	11.10 ± 0.14	0.62 ± 0.02
061222A	21.42 ± 0.22	2.29 ± 0.05	14.64	7.31 ± 0.77	0.27 ± 0.11
080721	21.85 ± 0.16	2.38 ± 0.03	17.75	6.15 ± 0.10	0.19 ± 0.01
100902A	⋯	⋯	⋯	8.09 ± 0.17	0.33 ± 0.01
111209A	⋯	⋯	⋯	5.88 ± 0.20	0.17 ± 0.01
140206A	22.53 ± 0.38	2.53 ± 0.09	14.81	7.60 ± 0.07	0.29 ± 0.01
150403A	19.21 ± 0.11	1.84 ± 0.02	5.77	16.62 ± 0.49	1.38 ± 0.09
	Silver
050315	16.26 ± 0.38	1.32 ± 0.06	6.87	11.83 ± 0.32	0.70 ± 0.04
050319	22.17 ± 0.96	2.45 ± 0.21	10.29	10.77 ± 0.59	0.58 ± 0.07
050505	15.14 ± 0.50	1.14 ± 0.08	9.10	8.31 ± 0.18	0.35 ± 0.01
050802	24.31 ± 0.71	2.94 ± 0.17	8.90	13.64 ± 0.51	0.93 ± 0.08
050803	33.30 ± 1.00	5.49 ± 0.33	6.26	26.57 ± 0.62	3.52 ± 0.17
060115	12.60 ± 1.69	0.79 ± 0.21	5.43	11.59 ± 0.58	0.67 ± 0.07
060210	21.96 ± 0.95	2.40 ± 0.21	12.20	8.99 ± 0.13	0.40 ± 0.01
060418	23.92 ± 1.07	2.85 ± 0.25	12.52	9.54 ± 0.30	0.46 ± 0.03
060502A	37.92 ± 1.21	7.10 ± 0.45	11.85	15.98 ± 0.24	1.28 ± 0.04
060604	20.74 ± 0.76	2.14 ± 0.16	3.95	26.25 ± 1.56	3.43 ± 0.50
060605	7.39 ± 0.25	0.27 ± 0.02	2.28	16.18 ± 0.40	1.31 ± 0.07
060614	57.78 ± 0.84	16.23 ± 0.46	6.93	41.67 ± 3.32	8.57 ± 2.38
060714	28.93 ± 1.14	4.16 ± 0.33	13.39	10.79 ± 0.53	0.58 ± 0.06
060729	81.80 ± 0.38	31.63 ± 0.28	18.69	21.86 ± 0.51	2.38 ± 0.12
060814	30.50 ± 0.66	4.61 ± 0.20	13.17	11.57 ± 0.21	0.67 ± 0.02
060906	11.94 ± 0.88	0.71 ± 0.10	5.73	10.41 ± 0.42	0.54 ± 0.05
060908	21.96 ± 0.83	2.40 ± 0.18	8.14	13.47 ± 0.16	0.91 ± 0.02
061121	34.03 ± 0.64	5.73 ± 0.21	23.64	7.19 ± 0.17	0.26 ± 0.01
070129	24.81 ± 0.69	3.06 ± 0.17	11.52	10.75 ± 0.29	0.58 ± 0.03
070306	21.11 ± 0.28	2.22 ± 0.06	8.65	12.20 ± 0.26	0.74 ± 0.04
070508	30.55 ± 0.42	4.63 ± 0.13	14.73	10.36 ± 0.20	0.54 ± 0.02
070529	17.39 ± 1.41	1.51 ± 0.24	7.02	12.37 ± 0.34	0.77 ± 0.04
080310	16.64 ± 0.47	1.38 ± 0.08	7.51	11.07 ± 0.75	0.61 ± 0.11
080430	49.60 ± 1.17	12.05 ± 0.56	10.86	22.81 ± 0.50	2.60 ± 0.12
080516	6.36 ± 0.57	0.20 ± 0.04	1.71	18.60 ± 0.87	1.73 ± 0.17
080605	16.90 ± 0.41	1.43 ± 0.07	8.69	9.72 ± 0.21	0.47 ± 0.02
080905B	23.01 ± 0.79	2.64 ± 0.18	9.80	11.73 ± 0.44	0.69 ± 0.05
081008	12.50 ± 1.05	0.78 ± 0.13	5.85	10.67 ± 0.40	0.57 ± 0.05
081221	20.59 ± 0.31	2.11 ± 0.06	13.94	7.38 ± 0.09	0.27 ± 0.01
090407	16.82 ± 0.78	1.41 ± 0.13	4.47	18.80 ± 1.04	1.77 ± 0.23
090418A	15.38 ± 0.37	1.18 ± 0.06	8.68	8.85 ± 0.31	0.39 ± 0.03
090510	11.78 ± 0.60	0.69 ± 0.07	2.40	24.56 ± 1.28	3.01 ± 0.35
090516	13.30 ± 0.37	0.88 ± 0.05	10.80	6.15 ± 0.17	0.19 ± 0.01
090529	17.12 ± 2.75	1.46 ± 0.47	5.06	16.90 ± 1.42	1.43 ± 0.32
090530	31.32 ± 1.35	4.86 ± 0.41	7.64	20.48 ± 0.64	2.09 ± 0.14
090618	49.17 ± 0.34	11.85 ± 0.16	26.06	9.42 ± 0.20	0.44 ± 0.02
091018	25.15 ± 0.57	3.14 ± 0.14	5.84	21.49 ± 0.64	2.31 ± 0.14
100418A	38.01 ± 1.29	7.14 ± 0.48	5.64	33.66 ± 5.25	5.62 ± 3.41
100615A	13.80 ± 0.38	0.95 ± 0.05	5.89	11.70 ± 0.13	0.69 ± 0.02
100621A	32.17 ± 1.89	5.13 ± 0.60	11.30	14.22 ± 0.39	1.01 ± 0.06
100704A	20.22 ± 0.48	2.04 ± 0.10	11.54	8.76 ± 0.14	0.38 ± 0.01
100814A	23.35 ± 0.46	2.71 ± 0.11	12.75	9.15 ± 0.15	0.42 ± 0.01
100906A	10.25 ± 0.29	0.53 ± 0.03	6.67	7.68 ± 0.07	0.29 ± 0.00
111008A	18.48 ± 0.29	1.70 ± 0.05	13.19	7.00 ± 0.16	0.25 ± 0.01
111123A	9.50 ± 0.62	0.45 ± 0.06	5.84	8.12 ± 0.13	0.33 ± 0.01
111228A	45.20 ± 0.88	10.04 ± 0.38	17.69	12.76 ± 0.35	0.81 ± 0.05
120327A	12.11 ± 0.65	0.73 ± 0.08	5.75	10.52 ± 0.11	0.55 ± 0.01
120712A	8.15 ± 0.40	0.33 ± 0.03	4.70	8.66 ± 0.03	0.38 ± 0.00
120811C	8.48 ± 0.72	0.36 ± 0.06	3.79	11.17 ± 0.44	0.62 ± 0.05
120922A	17.05 ± 1.19	1.45 ± 0.20	10.21	8.34 ± 0.34	0.35 ± 0.03
121024A	12.93 ± 1.65	0.83 ± 0.21	4.09	15.79 ± 0.49	1.25 ± 0.08
121128A	9.15 ± 0.21	0.42 ± 0.02	2.50	18.27 ± 0.51	1.67 ± 0.09
121211A	17.91 ± 1.84	1.60 ± 0.33	3.03	29.53 ± 0.89	4.34 ± 0.26
130420A	36.71 ± 1.83	6.66 ± 0.66	6.74	27.22 ± 8.91	3.69 ± 6.18
130606A	8.70 ± 0.88	0.38 ± 0.08	5.23	8.31 ± 0.20	0.35 ± 0.02
130612A	9.55 ± 1.24	0.46 ± 0.12	2.40	19.89 ± 0.75	1.98 ± 0.15
131030A	37.50 ± 1.36	6.95 ± 0.50	21.22	8.83 ± 0.08	0.39 ± 0.01
131105A	22.41 ± 1.02	2.50 ± 0.23	15.17	7.38 ± 0.15	0.27 ± 0.01
140512A	16.23 ± 0.43	1.31 ± 0.07	7.78	10.43 ± 0.12	0.54 ± 0.01
140518A	4.83 ± 0.27	0.12 ± 0.01	1.98	12.18 ± 0.58	0.74 ± 0.08
140703A	4.98 ± 0.23	0.12 ± 0.01	1.60	15.55 ± 0.23	1.21 ± 0.04
141121A	11.08 ± 0.74	0.61 ± 0.08	1.08	51.16 ± 2.00	12.83 ± 1.06
150910A	9.53 ± 0.20	0.45 ± 0.02	3.10	15.33 ± 0.67	1.18 ± 0.11
151027A	27.26 ± 0.19	3.69 ± 0.05	10.10	13.47 ± 0.33	0.91 ± 0.05
160121A	5.78 ± 1.13	0.17 ± 0.06	1.46	19.82 ± 1.65	1.96 ± 0.53
160227A	29.86 ± 0.58	4.42 ± 0.17	11.61	12.85 ± 1.40	0.83 ± 0.78
160327A	6.23 ± 0.35	0.19 ± 0.02	2.97	10.48 ± 1.80	0.55 ± 0.65
161117A	26.62 ± 0.72	3.52 ± 0.19	16.30	8.16 ± 0.07	0.33 ± 0.01
170113A	26.20 ± 0.58	3.41 ± 0.15	6.26	20.89 ± 3.28	2.18 ± 2.39
	Bronze
051109B	41.94 ± 1.24	8.67 ± 0.51	1.70	123.29 ± 5.00	66.97 ± 5.04
051221A	40.77 ± 1.01	8.19 ± 0.40	8.45	24.10 ± 3.51	2.90 ± 3.66
060526	⋯	⋯	⋯	16.32 ± 0.67	1.33 ± 0.13
060708	57.94 ± 2.16	16.32 ± 1.18	15.47	18.71 ± 0.53	1.75 ± 0.10
061021	⋯	⋯	⋯	22.15 ± 2.45	2.45 ± 1.30
061110A	44.19 ± 2.72	9.60 ± 1.16	9.65	22.88 ± 1.82	2.61 ± 0.63
061201	31.17 ± 2.04	4.82 ± 0.63	3.44	45.28 ± 10.32	10.10 ± 11.27
070110	⋯	⋯	⋯	13.94 ± 0.32	0.97 ± 0.04
080707	23.18 ± 0.81	2.67 ± 0.19	4.70	24.61 ± 0.91	3.02 ± 0.24
081007	49.07 ± 2.29	11.80 ± 1.08	6.50	37.69 ± 2.13	7.03 ± 0.89
081029	⋯	⋯	⋯	9.58 ± 0.38	0.46 ± 0.04
100219A	⋯	⋯	⋯	15.92 ± 0.80	1.27 ± 0.14
100302A	⋯	⋯	⋯	15.39 ± 0.71	1.18 ± 0.12
100425A	27.32 ± 1.49	3.71 ± 0.40	6.12	22.29 ± 1.41	2.48 ± 0.39
110715A	37.45 ± 0.52	6.93 ± 0.19	13.20	14.17 ± 0.17	1.00 ± 0.02
110808A	33.05 ± 2.64	5.41 ± 0.86	18.88	8.74 ± 0.39	0.38 ± 0.04
120422A	82.70 ± 7.70	32.29 ± 5.66	5.98	69.05 ± 3.36	22.95 ± 2.30
120521C	⋯	⋯	⋯	10.15 ± 0.34	0.52 ± 0.04
130408A	⋯	⋯	⋯	10.70 ± 0.55	0.57 ± 0.06
130603B	20.06 ± 1.08	2.00 ± 0.21	4.15	24.12 ± 0.81	2.90 ± 0.20
140903A	24.16 ± 1.01	2.90 ± 0.24	3.14	38.45 ± 1.67	7.31 ± 0.68
151215A	16.97 ± 1.96	1.44 ± 0.33	3.92	21.59 ± 4.87	2.33 ± 2.59
161108A	35.96 ± 1.65	6.40 ± 0.58	8.29	21.68 ± 1.33	2.35 ± 0.32

Notes.

^aEstimate of the jet-opening angle from the last data point of the Swift/XRT observation using Equation (16). ^ba_j was estimated by setting the ratio of θ_j,0 equal to θ_j. ^cEstimate of the jet-opening angle from the empirical relation of θ_j,0 ≈ 5.0/Γ₀ (Nemmen 2012).

Download table as:  ASCIITypeset images: 1 2

For a magnetized BH–accretion disk system, there is another magnetic mechanism: the magnetic coupling effect between the BH and the disk through closed magnetic field lines (van Putten 1999; Li & Paczyński 2000; Wang et al. 2002; Lei et al. 2009; Janiuk & Yuan 2010). Similar to the BZ mechanism, the magnetic coupling effect also extracts rotational energy from the spinning BH. Only if the BH spin is initially small would the magnetic coupling act as an additional spin-up process. A similar discussion of this aspect was made by Dai & Liu (2012) within the context of the magnetar central engine model. In more general cases, the magnetic coupling effect would not significantly affect the BH spin evolution (Lei et al. 2009).

A rapidly spinning magnetar model invokes a rapidly spinning, strongly magnetized neutron star called a "magnetar." There are two critical magnetar parameters for a magnetar undergoing bipolar spin-down: the initial spin period P₀ and the surface polar cap magnetic field B_p, which can be derived using the characteristic luminosity L₀ and the spin-down timescale τ. We derive the magnetar parameters P₀ and ${B}_{P}$ using Equations (23)–(27), assuming isotropic emission from the magnetar.

For a magnetar, the characteristic spin-down luminosity is:

$$\begin{eqnarray}&&{L}_{0}=1.0\times {10}^{49}\,\mathrm{erg}\,{{\rm{s}}}^{-1}({B}_{p,15}^{2}{P}_{0,-3}^{-4}{R}_{6}^{6}),\end{eqnarray} \tag{ 23 }$$

where R is the stellar radius.²³ In Equation (1), t is a measure of time in the rest frame, and τ is the initial spin-down timescale, which is defined by

$$\begin{eqnarray}&&\tau ={\displaystyle \frac{P}{2\dot{P}}}_{0}=2.05\times {10}^{3}s({I}_{45}{B}_{p,15}^{-2}{P}_{0,-3}^{2}{R}_{6}^{-6}),\end{eqnarray} \tag{ 24 }$$

where $\dot{P}$ is the rate of period increase due to MDR, and I₄₅ is the stellar moment of inertia in units of 10⁴⁵ g cm⁻² (Datta & Kapoor 1988; Weber & Glendenning 1992). The initial spin-down timescale, τ, in general, can be identified as the observed break time, t_b,

$$\begin{eqnarray}&&\tau ={t}_{{\rm{b}}},\end{eqnarray} \tag{ 25 }$$

and the characteristic spin-down luminosity L₀, in general, should include two terms,

$$\begin{eqnarray}&&{L}_{0}={L}_{{\rm{X}}}+{L}_{{\rm{K}}}=({L}_{{\rm{X}},\mathrm{iso}}+{L}_{{\rm{K}},\mathrm{iso}}),\end{eqnarray} \tag{ 26 }$$

where L_X,iso is the X-ray isotropic luminosity due to the internal dissipation of the magnetar wind, which is the observed X-ray luminosity of the internal plateau and measured at the observed break time (one can only derive an upper limit for external plateaus), and

$$\begin{eqnarray}&&{L}_{{\rm{K}},\mathrm{iso}}={E}_{{\rm{K}},\mathrm{iso}}(1+z)/{t}_{{\rm{b}}}\end{eqnarray} \tag{ 27 }$$

is the kinetic isotropic luminosity which is injected into the blast wave during the energy injection phase. It depends on the isotropic kinetic energy E_K,iso after the injection phase is over and the observed break time t_b. The isotropic kinetic energy ${E}_{{\rm{K}},\mathrm{iso}}$ can be derived from afterglow modeling, as discussed above.

Since it is not possible to completely rule out the magnetar progenitor for the Silver sample, we investigate the magnetar parameters for both the Silver and Bronze samples.

The derived values of P₀ and B_P are presented in Table 4 and Figure 12. We found that all bursts in the Bronze sample fall in a reasonable range in the P₀−B_P plot (Figure 12). B_P has a typical value of 10¹⁵ G and is in the range [10¹⁴ G, 10¹⁶ G]; there is no case with B_P below 10¹⁴ G, and only one case is greater than 10¹⁶ G. We find that most derived P₀ are close to 1 ms; this is consistent with the breakup limit of a neutron star, which is about 0.96 ms (Lattimer & Prakash 2004).²⁴

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Table 3.  The Properties of the BH Candidate (Gold+Silver) Samples

GRB	T90	Eγisoa	EK,iso(tb)b	${E}_{{\rm{K}},\mathrm{iso}}({t}_{0})$ b	a0c	p	q	z
	(s)	(1052 erg)	(1052 erg)	(1052 erg)	(10−1)			(redshift)
	Gold
050401	33.3	42.23 ± 8.95	71.53 ± 29.27	0.75 ± 0.50	1.39 ± 0.29	2.29 ± 0.06	−0.26 ± 0.14	2.90
050730	60.0	0.01 ± 0.00	...d	⋯	⋯	2.50 ± 0.10	1.06 ± 0.12	3.967
060607A	102.2	7.14 ± 0.34	⋯	⋯	⋯	2.50 ± 0.10	0.74 ± 0.12	3.0749
061222A	71.4	35.64 ± 14.53	158.72 ± 17.52	19.66 ± 6.52	2.06 ± 0.12	3.33 ± 0.03	0.61 ± 0.06	2.088
080721	16.2	69.31 ± 4.40	78.49 ± 6.24	30.10 ± 2.92	1.45 ± 0.06	3.20 ± 0.01	0.71 ± 0.02	2.591
100902A	428.8	24.09 ± 1.96	⋯	⋯	⋯	2.50 ± 0.10	0.27 ± 0.11	4.5
111209A	14,400.0	82.27 ± 10.88	⋯	⋯	⋯	2.50 ± 0.10	1.40 ± 0.01	0.677
140206A	93.6	30.68 ± 1.12	46.28 ± 8.34	21.92 ± 8.46	1.12 ± 0.10	2.88 ± 0.02	0.79 ± 0.10	2.73
150403A	40.9	1.51 ± 0.17	248.80 ± 15.64	4.02 ± 0.31	2.57 ± 0.08	2.26 ± 0.01	−0.21 ± 0.01	2.06
	Silver
050315	95.6	5.59 ± 0.58	388.63 ± 97.31	70.55 ± 24.44	3.19 ± 0.40	3.58 ± 0.16	0.55 ± 0.06	1.949
050319	152.5	8.02 ± 1.70	41.83 ± 19.26	2.86 ± 1.66	1.06 ± 0.25	3.08 ± 0.20	0.39 ± 0.07	3.240
050505	58.9	21.69 ± 1.78	224.36 ± 77.07	15.65 ± 6.21	2.44 ± 0.43	2.73 ± 0.06	−0.21 ± 0.06	4.27
050802	19.0	3.23 ± 0.47	16.33 ± 4.93	0.55 ± 0.21	0.67 ± 0.11	2.55 ± 0.04	−0.12 ± 0.07	1.71
050803	87.9	0.25 ± 0.02	37.50 ± 12.04	10.09 ± 6.43	1.01 ± 0.17	3.09 ± 0.06	0.64 ± 0.15	0.42
060115	139.6	6.05 ± 1.16	85.68 ± 119.93	26.30 ± 40.87	1.52 ± 0.67	2.61 ± 0.22	0.41 ± 0.27	3.53
060210	255.0	16.04 ± 0.89	42.78 ± 20.56	16.78 ± 8.86	1.08 ± 0.28	2.88 ± 0.05	0.53 ± 0.07	3.91
060418	103.1	12.75 ± 1.53	6.50 ± 3.16	4.02 ± 2.29	0.42 ± 0.10	3.10 ± 0.05	0.63 ± 0.21	1.489
060502A	28.4	1.76 ± 0.10	4.05 ± 1.39	0.32 ± 0.21	0.33 ± 0.06	2.59 ± 0.06	0.44 ± 0.12	1.51
060604	95.0	0.26 ± 0.06	29.57 ± 11.70	4.19 ± 3.01	0.89 ± 0.19	2.36 ± 0.06	0.32 ± 0.20	2.1357
060605	79.1	1.67 ± 0.16	145.59 ± 51.59	0.48 ± 0.22	1.97 ± 0.38	3.25 ± 0.08	−0.52 ± 0.07	3.78
060614	108.7	0.04 ± 0.01	8.33 ± 1.26	0.14 ± 0.03	0.48 ± 0.04	2.96 ± 0.04	−0.70 ± 0.06	0.125
060714	115.0	7.95 ± 1.50	5.39 ± 2.21	1.38 ± 0.81	0.38 ± 0.09	2.75 ± 0.05	0.44 ± 0.15	2.711
060729	115.3	0.53 ± 0.05	15.56 ± 0.76	0.19 ± 0.03	0.65 ± 0.02	2.90 ± 0.01	0.07 ± 0.03	0.54
060814	145.3	6.09 ± 0.42	32.66 ± 7.73	9.20 ± 4.16	0.94 ± 0.12	2.57 ± 0.03	0.51 ± 0.15	1.9229
060906	43.5	9.13 ± 1.42	28.23 ± 24.05	0.05 ± 0.05	0.87 ± 0.31	2.76 ± 0.17	−0.75 ± 0.11	3.686
060908	19.3	3.39 ± 0.15	4.86 ± 2.02	1.01 ± 0.67	0.36 ± 0.08	3.08 ± 0.07	0.27 ± 0.23	1.8836
061121	81.3	37.89 ± 3.43	28.81 ± 5.64	0.23 ± 0.07	0.88 ± 0.09	2.31 ± 0.02	−0.55 ± 0.07	1.314
070129	460.6	8.06 ± 0.83	42.92 ± 12.30	2.04 ± 1.24	1.08 ± 0.16	2.27 ± 0.04	−0.03 ± 0.17	2.3384
070306	209.5	4.96 ± 0.41	126.93 ± 17.89	5.76 ± 2.00	1.85 ± 0.14	3.51 ± 0.05	0.25 ± 0.08	1.4959
070508	20.9	9.29 ± 0.69	4.09 ± 0.63	0.18 ± 0.03	0.33 ± 0.02	2.24 ± 0.01	−0.33 ± 0.03	0.82
070529	109.2	4.70 ± 0.50	17.06 ± 15.08	17.91 ± 22.88	0.68 ± 0.24	2.39 ± 0.05	1.02 ± 0.39	2.4996
080310	365.0	7.20 ± 1.86	15.82 ± 4.29	0.34 ± 0.13	0.65 ± 0.09	2.50 ± 0.06	−0.85 ± 0.10	2.42
080430	16.2	0.45 ± 0.04	8.37 ± 2.10	0.39 ± 0.25	0.48 ± 0.06	2.20 ± 0.03	0.33 ± 0.13	0.767
080516	5.8	0.98 ± 0.18	35.50 ± 37.61	0.92 ± 1.82	0.98 ± 0.36	2.14 ± 0.12	−0.07 ± 0.47	3.2
080605	20.0	11.88 ± 1.01	6.90 ± 1.77	1.12 ± 0.33	0.43 ± 0.06	2.16 ± 0.02	−0.18 ± 0.04	1.6398
080905B	128.0	5.76 ± 0.83	16.44 ± 6.21	4.69 ± 3.24	0.67 ± 0.14	2.95 ± 0.04	0.59 ± 0.19	2.374
081008	185.5	8.30 ± 1.19	79.29 ± 72.06	1.16 ± 1.17	1.46 ± 0.51	2.80 ± 0.14	−0.18 ± 0.08	1.9685
081221	34.0	34.30 ± 1.55	8.03 ± 1.35	4.16 ± 0.88	0.47 ± 0.04	2.75 ± 0.02	0.18 ± 0.10	2.26
090407	310.0	0.94 ± 0.20	(6.90 ± 3.49)e2	5.13 ± 3.04	4.20 ± 1.04	2.94 ± 0.14	−0.58 ± 0.07	1.4485
090418A	56.0	17.06 ± 2.31	12.57 ± 2.99	1.45 ± 0.87	0.58 ± 0.07	3.10 ± 0.04	0.31 ± 0.17	1.608
090510	0.3	0.34 ± 0.07	8.24 ± 4.58	0.51 ± 0.31	0.47 ± 0.13	3.20 ± 0.09	−0.05 ± 0.08	0.903
090516	210.0	69.02 ± 7.54	75.01 ± 21.53	34.50 ± 11.42	1.42 ± 0.23	3.48 ± 0.08	0.47 ± 0.09	4.109
090529	100.0	1.41 ± 0.46	91.37 ± 155.70	0.62 ± 1.29	1.57 ± 0.77	2.65 ± 0.54	−0.41 ± 0.27	2.625
090530	48.0	0.68 ± 0.08	3.19 ± 1.40	0.14 ± 0.11	0.29 ± 0.07	2.73 ± 0.13	0.34 ± 0.12	1.266
090618	113.2	13.39 ± 1.09	12.72 ± 0.98	3.08 ± 0.43	0.59 ± 0.02	3.01 ± 0.01	0.53 ± 0.04	0.54
091018	4.4	0.56 ± 0.06	7.96 ± 1.96	2.91 ± 1.49	0.46 ± 0.06	2.33 ± 0.02	0.51 ± 0.22	0.971
100418A	7.0	0.10 ± 0.06	33.95 ± 11.78	(4.53 ± 4.21)e–4	0.96 ± 0.18	2.55 ± 0.10	−0.95 ± 0.14	0.6235
100615A	39.0	5.82 ± 0.24	16.66 ± 5.28	0.27 ± 0.18	0.67 ± 0.11	2.67 ± 0.10	0.10 ± 0.12	1.398
100621A	63.6	2.75 ± 0.29	91.35 ± 62.62	4.43 ± 4.08	1.57 ± 0.48	2.88 ± 0.08	0.45 ± 0.11	0.542
100704A	197.5	17.75 ± 1.07	29.45 ± 7.24	2.55 ± 1.12	0.89 ± 0.12	2.84 ± 0.05	0.42 ± 0.08	3.6
100814A	174.5	15.01 ± 0.97	(6.16 ± 1.25)e3	35.06 ± 8.51	9.68 ± 0.41	3.12 ± 0.06	−0.42 ± 0.03	1.44
100906A	114.4	29.48 ± 0.98	(3.33 ± 0.97)e2	3.49 ± 1.40	2.96 ± 0.42	3.16 ± 0.80	−0.20 ± 0.06	1.727
111008A	63.46	42.11 ± 3.66	23.77 ± 3.92	3.15 ± 0.88	0.80 ± 0.07	2.78 ± 0.03	0.25 ± 0.08	4.9898
111123A	290.0	23.78 ± 1.47	89.24 ± 71.83	20.07 ± 18.06	1.55 ± 0.53	3.70 ± 0.29	0.36 ± 0.15	3.1516
111228A	101.2	4.17 ± 0.45	13.89 ± 2.91	1.82 ± 0.90	0.61 ± 0.07	2.31 ± 0.02	0.21 ± 0.17	0.714
120327A	62.9	8.78 ± 0.34	8.45 ± 5.07	0.67 ± 0.46	0.48 ± 0.14	2.35 ± 0.06	−0.10 ± 0.11	2.813
120712A	14.7	18.57 ± 0.27	8.69 ± 4.74	1.77 ± 1.34	0.49 ± 0.14	3.19 ± 0.10	0.59 ± 0.13	4.1745
120811C	26.8	6.96 ± 1.04	27.80 ± 24.19	15.01 ± 15.21	0.87 ± 0.30	2.21 ± 0.09	0.65 ± 0.28	2.671
120922A	173.0	21.38 ± 3.36	59.62 ± 41.06	18.00 ± 16.35	1.27 ± 0.39	2.12 ± 0.04	0.21 ± 0.30	3.1
121024A	69.0	1.84 ± 0.22	32.61 ± 45.48	2.92 ± 4.39	0.94 ± 0.44	2.56 ± 0.18	−0.19 ± 0.12	2.298
121128A	23.3	1.05 ± 0.11	14.73 ± 3.61	4.40 ± 1.49	0.63 ± 0.08	3.24 ± 0.04	0.42 ± 0.10	2.2
121211A	182.0	0.17 ± 0.02	37.76 ± 37.17	24.19 ± 26.08	1.01 ± 0.37	2.68 ± 0.19	0.80 ± 0.19	1.023
130420A	123.5	0.23 ± 0.28	3.72 ± 1.86	0.24 ± 0.16	0.32 ± 0.08	2.69 ± 0.09	0.38 ± 0.08	1.297
130606A	276.58	21.67 ± 2.04	86.86 ± 92.64	25.01 ± 28.81	1.53 ± 0.59	2.76 ± 0.14	−0.39 ± 0.24	5.913
130612A	4.0	0.76 ± 0.11	3.19 ± 4.66	1.02 ± 1.75	0.29 ± 0.13	2.33 ± 0.16	0.17 ± 0.51	2.006
131030A	41.1	17.22 ± 0.60	6.73 ± 2.61	4.38 ± 1.80	0.43 ± 0.09	2.70 ± 0.02	0.81 ± 0.06	1.295
131105A	112.3	34.31 ± 2.63	22.24 ± 10.29	4.48 ± 3.53	0.78 ± 0.19	2.26 ± 0.05	0.35 ± 0.25	1.686
140512A	154.8	9.07 ± 0.41	51.44 ± 14.15	0.76 ± 0.28	1.18 ± 0.17	2.57 ± 0.05	0.03 ± 0.05	0.725
140518A	60.5	4.99 ± 0.92	8.41 ± 5.03	1.25 ± 0.86	0.48 ± 0.14	3.04 ± 0.18	0.14 ± 0.13	4.707
140703A	67.1	1.95 ± 0.11	(1.26 ± 0.58)e3	(2.53 ± 1.26)e2	5.53 ± 1.21	3.37 ± 0.10	−0.20 ± 0.10	3.14
141121A	549.9	0.02 ± 0.00	(5.42 ± 3.90)e4	(4.10 ± 3.31)e3	9.14 ± 2.98	3.63 ± 0.26	−0.39 ± 0.17	1.47
150910A	112.2	2.06 ± 0.35	(8.75 ± 2.03)e2	8.10 ± 4.27	4.69 ± 0.50	3.48 ± 0.06	−0.29 ± 0.13	1.359
151027A	129.69	3.38 ± 0.32	28.71 ± 2.04	3.51 ± 0.38	0.88 ± 0.03	3.24 ± 0.02	0.12 ± 0.03	0.81
160121A	12.0	0.77 ± 0.25	(2.08 ± 0.98)e2	8.52 ± 4.08	0.24 ± 0.07	3.18 ± 0.77	−0.68 ± 0.12	1.960
160227A	316.5	4.06 ± 1.70	12.76 ± 2.62	1.92 ± 0.77	0.59 ± 0.06	2.63 ± 0.03	0.38 ± 0.11	2.38
160327A	28.0	8.90 ± 5.88	10.38 ± 6.74	2.67 ± 2.49	0.53 ± 0.16	3.13 ± 0.14	0.51 ± 0.23	4.99
161117A	125.7	23.29 ± 0.76	42.15 ± 11.73	9.31 ± 17.07	1.07 ± 0.15	2.23 ± 0.03	0.23 ± 0.92	1.549
170113A	20.66	0.63 ± 0.38	8.07 ± 1.98	1.61 ± 0.70	0.47 ± 0.06	2.68 ± 0.03	0.53 ± 0.10	1.968

Notes.

^aWe searched for E_γ,iso in published papers or calculated it using the fluence and redshift extrapolated into 1–10⁴ keV (rest frame) with a spectral model and a k-correction, in units of 10⁵² erg. ^bThe kinetic energy, which is derived from the normal decay phase. The initial kinetic energy (upper limit) is estimated from the kinetic energy at the break time (t_b). ^cThe BH spin parameter a_•, in units of 10⁻¹ s. ^dWe did not calculate the kinetic energy for the bursts that have an internal plateau.

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However, we also found that approximately half of the Silver sample bursts have ${B}_{P}\lt {10}^{14}$ G. Moreover, most of the bursts have P₀ that is far below 1 ms. This result implies that the Silver sample bursts are more likely to have a BH central engine.

Using the derived value of P₀, the total rotation energy of the millisecond magnetar within this scenario is

$$\begin{eqnarray}&&{E}_{\mathrm{rot}}=\displaystyle \frac{1}{2}I{{\rm{\Omega }}}^{2}\simeq 2\times {10}^{52}\,\mathrm{erg}{M}_{1.4}{R}_{6}^{2}{P}_{0,-3}^{-2},\end{eqnarray} \tag{ 28 }$$

where I is the moment of inertia, Ω = 2Π/P₀ is the initial angular frequency of the neutron star, ${M}_{1.4}=M/1.4{M}_{\odot }$, and E_rot is the maximal total energy available and can be compared to the observed and inferred energies (see Section 3.3).

In conclusion, the properties of the bursts in the Bronze sample are consistent with the predictions of the magnetar central engine model.

In Figure 13, we compare the inferred E_γ,iso + E_K,iso with the total rotation energy E_rot of the BH (Silver; Section 3.1) and the millisecond magnetar (Bronze; Section 3.2). It is found that the GRBs are generally above and slightly above the E_rot = E_γ,iso + E_K,iso line. This is consistent with the hypothesis, namely, that all of the emission energy ultimately comes from the spin energy of the BH or magnetar.

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In Figure 14, we show the T₉₀ (rest frame) as a function of various energies for both the BH and magnetar candidate samples. All isotropic energies for most cases are above 2 × 10⁵² erg while all other energies for most cases with a beam correction are below 2 × 10⁵² erg.

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In Figure 15, we compare the histograms of the isotropic energies (${E}_{\gamma ,\mathrm{iso}}$, E_K,iso, E_X,iso) of the BH and magnetar candidate samples. The typical values of all kinds of energies (E_γ,iso, E_K,iso, E_X,iso) for the BH candidate samples (Gold+Silver) are systemically larger than those for the magnetar sample (Bronze).

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Figure 16 displays T₉₀ (rest frame) as a function of redshift for the samples. Two interesting findings: first, four out of the five short GRBs are in the magnetar candidate sample (Bronze); second, the redshifts of all five short bursts are below 1.

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Figure 17 shows the correlations of L_X,iso−E_γ,iso and L_X,iso−t_b/(1 + z) for the entire sample (BH candidate Gold+Silver samples and magnetar candidate Bronze sample). We find that a higher isotropic γ-ray energy implies a higher isotropic X-ray break luminosity, and the bursts of the Gold sample cluster into the region of high isotropic γ-ray energy and isotropic X-ray break luminosity. The best fit to the correlation is $\mathrm{log}{L}_{{\rm{X}},\mathrm{iso},49}=(-1.33\pm 0.10)+(1.06\,\pm 0.09)\mathrm{log}{E}_{\gamma ,\mathrm{iso},52}$, with a Spearman correlation coefficient of 0.78 and a chance probability of p < 10⁻⁴. For the L_X,iso–t_b/(1 + z) correlations, the best fit is log L_X,iso,49 =(2.74 ± 0.40) + (−1.09 ± 0.09) log t_b/(1 + z), with a Spearman correlation coefficient of 0.79 and a chance probability of p < 10⁻⁴. The anticorrelation between the luminosity and the rest-frame duration of the plateau phase is consistent with previous findings (Dainotti et al. 2008, 2010, 2013, 2015, 2017; Dall'Osso et al. 2011; Rowlinson et al. 2013, 2014, 2017).

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In Figure 18, we show the scatter plots of η_X versus η_γ(t_b), E_γ,iso, E_K,iso(t_b), and E_rot (Bronze). One interesting finding is η_X and E_K,iso(t_b) presenting an anticorrelation for the BH candidate samples (Gold+Silver) while presenting a positive correlation for the magnetar candidate sample (Bronze).

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Figure 19 displays the distributions of the temporal indices α in different subsamples and different temporal segments. They are all well fitted with Gaussian distributions for each sample. For the α_x,1 distributions, we have α_x,1 = 0.55 ± 0.30 for the BH candidate samples and α_x,1 = 0.32 ± 0.48 for the magnetar candidate sample. The same analysis based on the type of plateau gives α_x,1 = 0.60 ± 0.47 for the internal plateau sample and α_x,1 = 0.50 ± 0.31 for the external plateau ones (Figure 19(a)). For the α_x,2 distributions, we have log α_x,2 = 0.18 ± 0.11 for the BH candidate samples and log α_x,2 = 0.07 ± 0.13 for the magnetar candidate samples. For the internal/external plateau samples, one has log α_x,2 = 0.46 ± 0.33 for the internal plateau sample and log α_x,2 = 0.17 ± 0.11 for the external plateau sample (Figure 19(b)).

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Another self-consistency check is to compare the observed change of the decay slope (Δα_x = α_x,2 − α_x,1). We find log Δα_x = 0.01 ± 0.18 for the BH candidate samples and log Δα_x = 0.05 ± 0.39 for the magnetar candidate, and log Δα_x = 0.51 ± 0.60 for the internal plateau sample and log Δα_x = −0.01 ± 0.17 for external plateau sample (Figure 19(c)).

The temporal indices (both α₁ and α₂) for the BH candidate samples (Gold+Silver), in general, are steeper than those for the magnetar sample (Bronze). For the internal and external examples, the prebreak temporal indices α₁ are consistent with each other, but the postbreak temporal indices α₂ for the internal sample is much steeper than those for the external sample, consistent with an internal origin. The degrees of the break Δα for the BH candidate samples (Gold+Silver) is significantly less than those of the Silver and the magnetar samples (Bronze).

Figure 20 shows the distributions of the photon spectral indices Γ in different subsamples and different plateau types. They are all well-fitted with Gaussian distributions for each sample.

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For the Γ_x,1 distributions, we have Γ_x,1 = 1.91 ± 0.20 for the BH candidate samples and Γ_x,1 = 1.99 ± 0.23 for the magnetar candidate sample (Figure 20(a)). For the Γ_x,2 distributions, one has Γ_x,2 = 1.94 ± 0.20 for the BH candidate samples and Γ_x,2 = 1.88 ± 0.25 for the magnetar candidate sample (Figure 20(b)). For the ΔΓ_x distributions, one has ΔΓ_x = 0.03 ± 0.19 for the BH candidate samples and log ΔΓ_x = −0.08 ± 0.24 for the magnetar candidate sample (Figure 20(c)).

We also investigate the Γ distributions in different plateau types. For the internal plateau samples, one has Γ_x,1 = 2.00 ± 0.44, Γ_x,2 = 1.81 ± 0.53, and ΔΓ_x = 0.13 ±0.26, respectively. For the external plateau samples, one has Γ_x,1 = 1.92 ± 0.20, Γ_x,2 = 1.95 ± 0.17, and ΔΓ_x = 0.01 ±0.20, respectively.

The spectral indices (both Γ₁ and Γ₂) and the degrees of the break ΔΓ for the different subsamples (the BH candidate samples and the magnetar candidate sample), in general, present the same distributions. For the internal and external samples, a change is not found in the external sample, while a significant change in the internal sample is found.

The results indicate that there is no spectral evolution for the subsamples and the external plateau sample, while the internal plateau sample has a significant spectral change between the plateau phase and the following steeper decay phase. This is consistent with the internal plateau having an internal physical origin.

Figure 21 shows the distributions of the observed break times (t_b). A Gaussian fit to the distributions gives log (t_b/s) = (4.05 ± 0.68) and log (t_b/s) = (4.38 ± 0.63) for the BH candidate samples and the magnetar candidate sample, respectively. Separating the "internal" plateau and "external" plateau samples, we have log(t_b/s) = (4.22 ± 0.54) for the "internal" plateau sample and log(t_b/s) = (4.07 ± 0.72) for the "external" plateau sample. So, both external and internal plateau samples are consistent in terms of t_b.

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The typical break time for the BH candidate samples (Gold+Silver) are statistically earlier than that of the magnetar candidate sample (Bronze). Similar results can be found between the internal (earlier) and external (later) samples. The results indicate that the end energy injection for the "external plateau" is in general earlier than that for the "internal plateau" on average.

We derive the electron spectral indices p using temporal indices:

$$\begin{eqnarray}p=\left\{\begin{array}{ll}\tfrac{4{\alpha }_{2}+2}{3}, & \nu \gt \max ({\nu }_{m},{\nu }_{c})(\mathrm{ISM}\,\mathrm{and}\,\mathrm{wind})\\ \tfrac{4{\alpha }_{2}+3}{3}, & {\nu }_{m}\lt \nu \lt {\nu }_{c}(\mathrm{ISM})\\ \tfrac{4{\alpha }_{2}+1}{3}, & {\nu }_{m}\lt \nu \lt {\nu }_{c}(\mathrm{wind}).\end{array}\right.\end{eqnarray} \tag{ 29 }$$

The p distributions for our subsamples are p = 2.63 ± 0.63 for the BH candidate samples and p = 2.49 ± 0.15 for the magnetar candidate sample (Figure 22). The global p value is within the range [1.89, 3.70], and it has a Gaussian distribution with a center value p = 2.69 ± 0.05. Except for 11 GRBs that have internal plateaus, all the rest have p > 2.0, except for one case (GRB 100302A) which had p < 2.0 (Figure 22).

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The results are consistent with the typical value of p for relativistic shocks due to first-order Fermi acceleration (e.g., Achterberg et al. 2001; Ellison & Double 2002).

Similarly, the luminosity injection index q (see Equation (1)) can be determined from the temporal index α and the spectral index β for different afterglow models; here, α and β are measured in the shallow decay segment preceeding the break:

$$\begin{eqnarray}q=\left\{\begin{array}{ll}\tfrac{2{\alpha }_{1}-2{\beta }_{1}+2}{1+{\beta }_{1}}, & \nu \gt \max ({\nu }_{m},{\nu }_{c}){\rm{(ISM\; and\; wind)}}\\ \tfrac{2{\alpha }_{1}-2{\beta }_{1}+2}{2+{\beta }_{1}}, & {\nu }_{m}\lt \nu \lt {\nu }_{c}{\rm{(ISM)}}\\ \tfrac{2{\alpha }_{1}-2{\beta }_{1}}{1+{\beta }_{1}}, & {\nu }_{m}\lt \nu \lt {\nu }_{c}{\rm{(wind)}}.\end{array}\right.\end{eqnarray} \tag{ 30 }$$

The q distributions for our subsamples are q = 0.24 ± 0.50 for the BH candidate samples, and q = 0.38 ± 0.35 for the magnetar candidate sample (Figure 23). Also, for the internal plateau, we have q = 0.39 ± 0.35, and for the external plateau, we have q = 0.22 ± 0.35.

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The different subsamples present different central values; the typical value for the BH candidate samples (Gold+Silver) is slightly less than that for the magnetar candidate sample (Bronze). The typical q value for the internal plateau sample is greater than that for the external plateau sample. In total, the central q value for the subsamples is around 0.3, which is consistent with the predictions of an energy injection model (Zhang et al. 2006).

One important assumption in the analysis above is whether or not the emitted energy is isotropic, since this affects the derived energetics of the bursts. In the framework of a BH central engine, the rotational energy can naturally be assumed to be extracted through the jet and deposited into the X-ray plateau. In this case, a beaming correction needs to be made: E_K = f_bE_K,iso. On the other hand, for a magnetar central engine, the rotational energy is initially emitted in an isotropic wind. Indeed, for short GRBs, the total energy is typically expected to be emitted quasi-spherically (e.g., Fan & Xu 2006; Metzger & Piro 2014). Here, we have also assumed the energy to be emitted quasi-isotropically even for long GRBs during their X-ray afterglow phase (see, e.g., Mao et al. 2010; Dall'Osso et al. 2011; Mazzali et al. 2014). Then only a small part of the rotational energy of a magnetar is injected into the external shock of the jet. This means that for the magnetar sample, using this assumption, we should not use the beaming correction, so that f_b = 1.

We note, however, that it is still unclear whether the energy release for long GRBs during their X-ray afterglow phase is spherical or collimated (Metzger et al. 2011), and there needs to be further theoretical and numerical modeling, as well as observational constraints, to understand the late-time energy release from a magnetar. Indeed, there are arguments for a collimated outflow from a magnetar central engine. For instance, the progenitor star of long GRBs is expected to cause a collimation of the emitted energy, at least during the prompt phase (Bucciantini et al. 2007, 2009), and this would change the derived total energetics. However, for very late times, such as during the X-ray plateau phase, which is a hundred seconds to a day following the trigger of the GRB, there should be less material surrounding the magnetar. Therefore, the magnetar emission could be expected to come out more isotropically.

Similarly, Lü & Zhang (2014) found, using the Swift data of GRB afterglows, that while an isotropic assumption for the emitted energy is reasonable for short GRBs, for most long GRBs, this assumption leads to unrealistic values of the derived rotation period of the neutron star. Lü & Zhang (2014) therefore argue that this means that the assumption of isotropic energy release during the afterglow phase is not correct. We note, however, that it could also point to to the fact that the progenitor is not a magnetar.

By contrast, an indirect argument for quasi-isotropic magnetar winds is given in Mazzali et al. (2014) who studied the energetics of GRBs associated with SNe. They noted that the SN energy is narrowly distributed around the characteristic magnetar energy of ~10⁵² erg and is always larger than the GRB energy. On the other hand, the GRB energy varies a lot. They therefore suggest that the central engine for the SN-associated GRBs is a magnetar and that the SN is driven by a close-to-spherical magnetar wind. A consequence is that the jet does not drive the stellar explosion; indeed, it is not clear how the jet would transfer energy to the star after it has escaped. Mazzali et al. (2014) had to make an assumption, though, that the magnetar wind is not fully isotropic based on the fact that SNe, in general, are observed to have some degree of asphericity (e.g., Maeda et al. 2002; Tanaka et al. 2007). The asphericity is, however, assumed to be much smaller than the highly collimated jetted flows from BH central engines, with only f_b < ~0.5−0.2. Moreover, Greiner et al. (2015) argued for a magnetar central engine in GRB 111209A, since the associated SN 2011kl did not show any evidence for Ni production. However, its properties are peculiar compared to other GRB/SN cases and might thus not be representative (Cano et al. 2017). In the case of GRB 111209A, a correction for collimation of ${f}_{{\rm{b}}}\lt 1/50$ was required for E_γ, which would suggest an initial jetted outflow. On the other hand, considering the gamma-ray radiative efficiency found in Section 2.6 (Equation (18), Figures 9 and 10) and the derived E_X,iso, the total energy budget could be strongly strained for simple estimates of a magnetar central engine, in particular if E_X is assumed to be isotropic. Indeed, Cano (2016) argues that additional energy sources, such as radioactive heating, have to be considered apart from the magnetar itself for a consistent scenario.

Following these lines of argument, the assumption made in this paper of f_b = 1 might be too high. Instead, by employing a factor of f_b <  0.5−0.2, as argued by Mazzali et al. (2014), the energies become smaller, and more bursts are consistent with being magnetar-bound. For instance, using f_b = 0.5 (0.2), there are 2 (18) bursts in the Silver sample that would be classified as possibly harboring a magnetar central engine and thus belonging to the Bronze sample (see Table 5). We also calculate the beaming correction, f_b, that individual bursts need in order to reach the magnetar limit (green line in Figure 8(a)). These f_b values are significantly larger than the f_b values obtained from the jet break. On the other hand, it shows that only a slight anisotropy is needed in order for the total energy to get below the magnetar limit. In such cases, the central engine could still be a magnetar.

This points toward the fact that there could be a continuum of degrees of asphericity (or collimation) of the magnetar wind. By assuming a fully isotropic wind as we did in this paper, then most bursts are inconsistent with the magnetar central engine (our Gold and Silver samples). Only a small fraction of bursts, both short GRBs and long GRBs in our Bronze sample, are consistent with a magnetar central engine assumption. By contrast, Lü & Zhang (2014) assume a high degree of collimation and find a large fraction of all bursts to be consistent with the magnetar central engine. As argued above, the degree of asphericity could lie in between the assumption of Lü & Zhang (2014; high degree of collimation) and this paper (isotropic wind) and could, for instance, be determined by varying the mass of the envelope surrounding the magnetar, which would have a different impact from the focusing of the wind.

In this paper, we systematically analyzed the Swift/XRT light curves having a plateau phase in 101 GRBs that were detected before 2017 May. Assuming an isotropic magnetar wind, we draw the following conclusions:

• 1.  
  A Gold sample, which includes 9 out of the 101 GRBs. Both E_X,iso and E_K,iso exceed the energy budget of a magnetar. Therefore, these most likely harbor a BH central engine.
• 2.  
  A Silver sample, which includes 69 out of the 101 GRBs. Estimates of their kinetic energy E_K,iso are larger than the energy budget of a magnetar. Likewise, these most likely have a BH central engine.
• 3.  
  A Bronze sample, which includes 23 out of the 101 GRBs. These bursts have an energy output lower than the maximal energy budget of magnetars, and as such can be considered candidates for the magnetar central engine.
• 4.  
  We further test the data with the BZ mechanism based on the BH model, and find that the observations of the "Gold" and "Silver" samples are consistent with expectations. We also test the magnetar model for the Silver and Bronze samples and find that the magnetar surface magnetic field (B_p) and initial spin period (P₀) fall into a reasonable range for the Bronze sample but not for the Silver sample. This implies that the Bronze sample is consistent with having a magnetar central engine, but the Silver sample, in general, is inconsistent with the magnetar model.

We also compare the properties of the two central engine candidate samples, and the two types of plateau bursts.

• 1.  
  BH candidate (Gold+Silver) and magnetar candidate (Bronze) samples. We found that the observational properties (α₁, α₂, Δα, Γ₁, Γ₂, and ΔΓ) and the model parameters (q), in general, do not differ significantly. This indicates that it could be insensitive to the central engine, or in the case where these parameters were to be sensitive to the central engine, it would argue for a common central engine type. However, one interesting result is that the p value distribution for the magnetar candidate sample is significantly narrower than that for the BH candidate sample. A significant difference is also found for t_b, z, T₉₀, L_X,iso, energies (E_X,iso, E_K,iso, and E_γ,iso), and p, which could imply different physical reasons for the different samples.
• 2.  
  Internal plateau and external plateau samples. For the internal and external plateaus, we found that α₂, Δα, Γ₂, ΔΓ, t_b, and q, in general, present different distributions. The results are consistent with the expectation for different physical progenitors for the two types of plateaus, while the α₁ and Γ₁ generally show similar distributions.

We conclude that, under the assumption of isotropic energy release from a magnetar, most GRBs should harbor a BH central engine.